diff --git a/synapse/storage/data_stores/main/events_bg_updates.py b/synapse/storage/data_stores/main/events_bg_updates.py
index efb9cd57af8dd2426110acbf267a6812e185b0a9..5177b71016000ac177db70365ed90cab832b2f27 100644
--- a/synapse/storage/data_stores/main/events_bg_updates.py
+++ b/synapse/storage/data_stores/main/events_bg_updates.py
@@ -90,14 +90,6 @@ class EventsBackgroundUpdatesStore(SQLBaseStore):
             "event_store_labels", self._event_store_labels
         )
 
-        self.db.updates.register_background_index_update(
-            "redactions_have_censored_idx",
-            index_name="redactions_have_censored",
-            table="redactions",
-            columns=["event_id"],
-            where_clause="NOT have_censored",
-        )
-
         self.db.updates.register_background_index_update(
             "redactions_have_censored_ts_idx",
             index_name="redactions_have_censored_ts",
diff --git a/synapse/storage/data_stores/main/schema/delta/56/redaction_censor.sql b/synapse/storage/data_stores/main/schema/delta/56/redaction_censor.sql
index a8583b52ccbd711941d884c86101a46740c1764f..ea95db0ed717b5ce51c23235875145a062e409a2 100644
--- a/synapse/storage/data_stores/main/schema/delta/56/redaction_censor.sql
+++ b/synapse/storage/data_stores/main/schema/delta/56/redaction_censor.sql
@@ -14,6 +14,3 @@
  */
 
 ALTER TABLE redactions ADD COLUMN have_censored BOOL NOT NULL DEFAULT false;
-
-INSERT INTO background_updates (update_name, progress_json) VALUES
-  ('redactions_have_censored_idx', '{}');
diff --git a/synapse/storage/data_stores/main/schema/delta/56/redaction_censor2.sql b/synapse/storage/data_stores/main/schema/delta/56/redaction_censor2.sql
index 6c36bd5468e977e12dac99f0e235a7e886bda4e4..49ce35d7947258126519c961e75ae2a035f724b9 100644
--- a/synapse/storage/data_stores/main/schema/delta/56/redaction_censor2.sql
+++ b/synapse/storage/data_stores/main/schema/delta/56/redaction_censor2.sql
@@ -18,5 +18,5 @@ ALTER TABLE redactions ADD COLUMN received_ts BIGINT;
 INSERT INTO background_updates (update_name, progress_json) VALUES
   ('redactions_received_ts', '{}');
 
-INSERT INTO background_updates (update_name, progress_json, depends_on) VALUES
-  ('redactions_have_censored_ts_idx', '{}', 'redactions_have_censored_idx');
+INSERT INTO background_updates (update_name, progress_json) VALUES
+  ('redactions_have_censored_ts_idx', '{}');
diff --git a/synapse/storage/data_stores/main/schema/delta/56/redaction_censor4.sql b/synapse/storage/data_stores/main/schema/delta/56/redaction_censor4.sql
new file mode 100644
index 0000000000000000000000000000000000000000..b7550f6f4e7082f5f6de0022fcd5d58cbdf93a9c
--- /dev/null
+++ b/synapse/storage/data_stores/main/schema/delta/56/redaction_censor4.sql
@@ -0,0 +1,16 @@
+/* Copyright 2019 The Matrix.org Foundation C.I.C.
+ *
+ * Licensed under the Apache License, Version 2.0 (the "License");
+ * you may not use this file except in compliance with the License.
+ * You may obtain a copy of the License at
+ *
+ *    http://www.apache.org/licenses/LICENSE-2.0
+ *
+ * Unless required by applicable law or agreed to in writing, software
+ * distributed under the License is distributed on an "AS IS" BASIS,
+ * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
+ * See the License for the specific language governing permissions and
+ * limitations under the License.
+ */
+
+DROP INDEX IF EXISTS redactions_have_censored;