diff --git a/oak-core/src/main/java/org/apache/jackrabbit/oak/plugins/document/rdb/RDBDocumentStoreJDBC.java b/oak-core/src/main/java/org/apache/jackrabbit/oak/plugins/document/rdb/RDBDocumentStoreJDBC.java
index 1004afa..a737d2a 100644
--- a/oak-core/src/main/java/org/apache/jackrabbit/oak/plugins/document/rdb/RDBDocumentStoreJDBC.java
+++ b/oak-core/src/main/java/org/apache/jackrabbit/oak/plugins/document/rdb/RDBDocumentStoreJDBC.java
@@ -302,7 +302,13 @@ public class RDBDocumentStoreJDBC {
             results = stmt.executeBatch();
         } catch (BatchUpdateException ex) {
             LOG.debug("Some of the batch updates failed", ex);
-            results = ex.getUpdateCounts();
+            // POSTGRES returns positive update counts even for rows that wasn't created.
+            // See OAK-3937.
+            if (this.dbInfo == RDBDocumentStoreDB.POSTGRES) {
+                results = new int[0];
+            } else {
+                results = ex.getUpdateCounts();
+            }
         } finally {
             stmt.close();
         }
diff --git a/oak-core/src/test/java/org/apache/jackrabbit/oak/plugins/document/MultiDocumentStoreTest.java b/oak-core/src/test/java/org/apache/jackrabbit/oak/plugins/document/MultiDocumentStoreTest.java
index 360cd3e..c041d6f 100644
--- a/oak-core/src/test/java/org/apache/jackrabbit/oak/plugins/document/MultiDocumentStoreTest.java
+++ b/oak-core/src/test/java/org/apache/jackrabbit/oak/plugins/document/MultiDocumentStoreTest.java
@@ -215,8 +215,6 @@ public class MultiDocumentStoreTest extends AbstractMultiDocumentStoreTest {
 
     @Test
     public void concurrentBatchUpdate() throws Exception {
-        // OAK-3924 and OAK-3937
-        assumeTrue(dsf == DocumentStoreFixture.MONGO);
         final CountDownLatch ready = new CountDownLatch(2);
         final CountDownLatch go = new CountDownLatch(1);
         final List<String> ids = Lists.newArrayList();