diff --git src/wp-includes/class-wp-customize-manager.php src/wp-includes/class-wp-customize-manager.php
index 440a537..085a204 100644
--- src/wp-includes/class-wp-customize-manager.php
+++ src/wp-includes/class-wp-customize-manager.php
@@ -1006,7 +1006,11 @@ final class WP_Customize_Manager {
 				'posts_per_page' => -1,
 			) );
 			foreach ( $existing_posts_query->posts as $existing_post ) {
-				$existing_starter_content_posts[ $existing_post->post_type . ':' . $existing_post->post_name ] = $existing_post;
+				$post_name = $existing_post->post_name;
+				if ( empty( $post_name ) ) {
+					$post_name = get_post_meta( $existing_post->ID, '_customize_draft_post_name', true );
+				}
+				$existing_starter_content_posts[ $existing_post->post_type . ':' . $post_name ] = $existing_post;
 			}
 		}
 
diff --git src/wp-includes/class-wp-customize-nav-menus.php src/wp-includes/class-wp-customize-nav-menus.php
index 1305a14..ca1b72f 100644
--- src/wp-includes/class-wp-customize-nav-menus.php
+++ src/wp-includes/class-wp-customize-nav-menus.php
@@ -803,6 +803,11 @@ final class WP_Customize_Nav_Menus {
 		if ( empty( $postarr['post_name'] ) ) {
 			$postarr['post_name'] = sanitize_title( $postarr['post_title'] );
 		}
+		if ( ! isset( $postarr['meta_input'] ) ) {
+			$postarr['meta_input'] = array();
+		}
+		$postarr['meta_input']['_customize_draft_post_name'] = $postarr['post_name'];
+		unset( $postarr['post_name'] );
 
 		add_filter( 'wp_insert_post_empty_content', '__return_false', 1000 );
 		$r = wp_insert_post( wp_slash( $postarr ), true );
@@ -1192,9 +1197,19 @@ final class WP_Customize_Nav_Menus {
 		if ( ! empty( $post_ids ) ) {
 			foreach ( $post_ids as $post_id ) {
 				$target_status = 'attachment' === get_post_type( $post_id ) ? 'inherit' : 'publish';
+				$args = array(
+					'ID' => $post_id,
+					'post_status' => $target_status,
+				);
+				$post_name = get_post_meta( $post_id, '_customize_draft_post_name', true );
+				if ( $post_name ) {
+					$args['post_name'] = $post_name;
+				}
 
 				// Note that wp_publish_post() cannot be used because unique slugs need to be assigned.
-				wp_update_post( array( 'ID' => $post_id, 'post_status' => $target_status ) );
+				wp_update_post( wp_slash( $args ) );
+
+				delete_post_meta( $post_id, '_customize_draft_post_name' );
 			}
 		}
 	}
diff --git src/wp-includes/post.php src/wp-includes/post.php
index 1496d6d..c81f5c5 100644
--- src/wp-includes/post.php
+++ src/wp-includes/post.php
@@ -3678,7 +3678,7 @@ function wp_unique_post_slug( $slug, $post_ID, $post_status, $post_type, $post_p
 
 	if ( 'attachment' == $post_type ) {
 		// Attachment slugs must be unique across all types.
-		$check_sql = "SELECT post_name FROM $wpdb->posts WHERE post_status != 'auto-draft' AND post_name = %s AND ID != %d LIMIT 1";
+		$check_sql = "SELECT post_name FROM $wpdb->posts WHERE post_name = %s AND ID != %d LIMIT 1";
 		$post_name_check = $wpdb->get_var( $wpdb->prepare( $check_sql, $slug, $post_ID ) );
 
 		/**
@@ -3706,7 +3706,7 @@ function wp_unique_post_slug( $slug, $post_ID, $post_status, $post_type, $post_p
 		 * Page slugs must be unique within their own trees. Pages are in a separate
 		 * namespace than posts so page slugs are allowed to overlap post slugs.
 		 */
-		$check_sql = "SELECT post_name FROM $wpdb->posts WHERE post_status != 'auto-draft' AND post_name = %s AND post_type IN ( %s, 'attachment' ) AND ID != %d AND post_parent = %d LIMIT 1";
+		$check_sql = "SELECT post_name FROM $wpdb->posts WHERE post_name = %s AND post_type IN ( %s, 'attachment' ) AND ID != %d AND post_parent = %d LIMIT 1";
 		$post_name_check = $wpdb->get_var( $wpdb->prepare( $check_sql, $slug, $post_type, $post_ID, $post_parent ) );
 
 		/**
@@ -3730,7 +3730,7 @@ function wp_unique_post_slug( $slug, $post_ID, $post_status, $post_type, $post_p
 		}
 	} else {
 		// Post slugs must be unique across all posts.
-		$check_sql = "SELECT post_name FROM $wpdb->posts WHERE post_status != 'auto-draft' AND post_name = %s AND post_type = %s AND ID != %d LIMIT 1";
+		$check_sql = "SELECT post_name FROM $wpdb->posts WHERE post_name = %s AND post_type = %s AND ID != %d LIMIT 1";
 		$post_name_check = $wpdb->get_var( $wpdb->prepare( $check_sql, $slug, $post_type, $post_ID ) );
 
 		// Prevent new post slugs that could result in URLs that conflict with date archives.
diff --git tests/phpunit/tests/customize/manager.php tests/phpunit/tests/customize/manager.php
index 345e666..84eb3d4 100644
--- tests/phpunit/tests/customize/manager.php
+++ tests/phpunit/tests/customize/manager.php
@@ -454,7 +454,11 @@ class Tests_WP_Customize_Manager extends WP_UnitTestCase {
 			} else {
 				$this->assertEquals( 'auto-draft', $post->post_status );
 			}
-			$posts_by_name[ $post->post_name ] = $post->ID;
+			$post_name = $post->post_name;
+			if ( empty( $post_name ) ) {
+				$post_name = get_post_meta( $post->ID, '_customize_draft_post_name', true );
+			}
+			$posts_by_name[ $post_name ] = $post->ID;
 		}
 		$this->assertEquals( array( 'waffles', 'canola', 'home', 'about', 'blog', 'custom' ), array_keys( $posts_by_name ) );
 		$this->assertEquals( 'Custom', get_post( $posts_by_name['custom'] )->post_title );
diff --git tests/phpunit/tests/customize/nav-menus.php tests/phpunit/tests/customize/nav-menus.php
index f8b38a1..e68ab05 100644
--- tests/phpunit/tests/customize/nav-menus.php
+++ tests/phpunit/tests/customize/nav-menus.php
@@ -549,14 +549,16 @@ class Test_WP_Customize_Nav_Menus extends WP_UnitTestCase {
 		$r = $menus->insert_auto_draft_post( array( 'post_title' => 'Hello World', 'post_type' => 'post' ) );
 		$this->assertInstanceOf( 'WP_Post', $r );
 		$this->assertEquals( 'Hello World', $r->post_title );
-		$this->assertEquals( 'hello-world', $r->post_name );
+		$this->assertEquals( '', $r->post_name );
+		$this->assertEquals( 'hello-world', get_post_meta( $r->ID, '_customize_draft_post_name', true ) );
 		$this->assertEquals( 'post', $r->post_type );
 
 		$r = $menus->insert_auto_draft_post( array( 'post_title' => 'Hello World', 'post_type' => 'post', 'post_name' => 'greetings-world', 'post_content' => 'Hi World' ) );
 		$this->assertInstanceOf( 'WP_Post', $r );
 		$this->assertEquals( 'Hello World', $r->post_title );
 		$this->assertEquals( 'post', $r->post_type );
-		$this->assertEquals( 'greetings-world', $r->post_name );
+		$this->assertEquals( '', $r->post_name );
+		$this->assertEquals( 'greetings-world', get_post_meta( $r->ID, '_customize_draft_post_name', true ) );
 		$this->assertEquals( 'Hi World', $r->post_content );
 	}
 
@@ -739,11 +741,17 @@ class Test_WP_Customize_Nav_Menus extends WP_UnitTestCase {
 		$menus = new WP_Customize_Nav_Menus( $this->wp_customize );
 		do_action( 'customize_register', $this->wp_customize );
 
-		$post_ids = $this->factory()->post->create_many( 3, array(
-			'post_status' => 'auto-draft',
-			'post_type' => 'post',
-			'post_name' => 'auto-draft',
-		) );
+		$post_ids = array();
+		for ( $i = 0; $i < 3; $i += 1 ) {
+			$r = $menus->insert_auto_draft_post( array(
+				'post_title' => 'Auto Draft ' . $i,
+				'post_type' => 'post',
+				'post_name' => 'auto-draft-' . $i,
+			) );
+			$this->assertInstanceOf( 'WP_Post', $r );
+			$post_ids[] = $r->ID;
+		}
+
 		$pre_published_post_id = $this->factory()->post->create( array( 'post_status' => 'publish' ) );
 
 		$setting_id = 'nav_menus_created_posts';
@@ -754,6 +762,8 @@ class Test_WP_Customize_Nav_Menus extends WP_UnitTestCase {
 		$this->assertEquals( $post_ids, $setting->post_value() );
 		foreach ( $post_ids as $post_id ) {
 			$this->assertEquals( 'auto-draft', get_post_status( $post_id ) );
+			$this->assertEmpty( get_post( $post_id )->post_name );
+			$this->assertNotEmpty( get_post_meta( $post_id, '_customize_draft_post_name', true ) );
 		}
 
 		$save_action_count = did_action( 'customize_save_nav_menus_created_posts' );
@@ -761,6 +771,8 @@ class Test_WP_Customize_Nav_Menus extends WP_UnitTestCase {
 		$this->assertEquals( $save_action_count + 1, did_action( 'customize_save_nav_menus_created_posts' ) );
 		foreach ( $post_ids as $post_id ) {
 			$this->assertEquals( 'publish', get_post_status( $post_id ) );
+			$this->assertRegExp( '/^auto-draft-\d+$/', get_post( $post_id )->post_name );
+			$this->assertEmpty( get_post_meta( $post_id, '_customize_draft_post_name', true ) );
 		}
 
 		// Ensure that unique slugs were assigned.
diff --git tests/phpunit/tests/post/wpUniquePostSlug.php tests/phpunit/tests/post/wpUniquePostSlug.php
index a728454..536017d 100644
--- tests/phpunit/tests/post/wpUniquePostSlug.php
+++ tests/phpunit/tests/post/wpUniquePostSlug.php
@@ -347,46 +347,4 @@ class Tests_Post_WpUniquePostSlug extends WP_UnitTestCase {
 		$found = wp_unique_post_slug( 'embed', $p, 'publish', 'attachment', 0 );
 		$this->assertSame( 'embed-2', $found );
 	}
-
-	/**
-	 * @ticket 38928
-	 */
-	public function test_non_unique_slugs_for_existing_auto_draft_posts() {
-		$auto_draft_post_id = self::factory()->post->create( array(
-			'post_type' => 'post',
-			'post_name' => 'existing-post',
-			'post_status' => 'auto-draft',
-		) );
-		$auto_draft_page_id = self::factory()->post->create( array(
-			'post_type' => 'page',
-			'post_name' => 'existing-page',
-			'post_status' => 'auto-draft',
-		) );
-		$auto_draft_attachment_id = self::factory()->attachment->create_object( 'image.jpg', $auto_draft_page_id, array(
-			'post_mime_type' => 'image/jpeg',
-			'post_type' => 'attachment',
-			'post_name' => 'existing-attachment',
-			'post_status' => 'auto-draft',
-		) );
-
-		$post_id = self::factory()->post->create( array( 'post_type' => 'post' ) );
-		$page_id = self::factory()->post->create( array( 'post_type' => 'page' ) );
-		$attachment_id = self::factory()->attachment->create_object( 'image2.jpg', $page_id, array(
-			'post_mime_type' => 'image/jpeg',
-			'post_type' => 'attachment',
-			'post_name' => 'existing-image',
-		) );
-
-		$this->assertEquals( 'existing-post', wp_unique_post_slug( 'existing-post', $post_id, 'publish', get_post_type( $post_id ), 0 ) );
-		wp_publish_post( $auto_draft_post_id );
-		$this->assertEquals( 'existing-post-2', wp_unique_post_slug( 'existing-post', $post_id, 'publish', get_post_type( $post_id ), 0 ) );
-
-		$this->assertEquals( 'existing-page', wp_unique_post_slug( 'existing-page', $page_id, 'publish', get_post_type( $page_id ), 0 ) );
-		wp_publish_post( $auto_draft_page_id );
-		$this->assertEquals( 'existing-page-2', wp_unique_post_slug( 'existing-page', $page_id, 'publish', get_post_type( $page_id ), 0 ) );
-
-		$this->assertEquals( 'existing-attachment', wp_unique_post_slug( 'existing-attachment', $attachment_id, 'publish', get_post_type( $attachment_id ), 0 ) );
-		wp_publish_post( $auto_draft_attachment_id );
-		$this->assertEquals( 'existing-attachment-2', wp_unique_post_slug( 'existing-attachment', $attachment_id, 'publish', get_post_type( $attachment_id ), 0 ) );
-	}
 }