diff --git a/requirements.txt b/requirements.txt
--- a/requirements.txt
+++ b/requirements.txt
@@ -1,3 +1,5 @@
 # Add here external Python modules dependencies, one per line. Module names
 # should match https://pypi.python.org/pypi names. For the full spec or
 # dependency lines, see https://pip.readthedocs.org/en/1.1/requirements.html
+
+tenacity >= 6.2
diff --git a/swh/objstorage/replayer/replay.py b/swh/objstorage/replayer/replay.py
--- a/swh/objstorage/replayer/replay.py
+++ b/swh/objstorage/replayer/replay.py
@@ -1,4 +1,4 @@
-# Copyright (C) 2019-2020 The Software Heritage developers
+# Copyright (C) 2019-2021 The Software Heritage developers
 # See the AUTHORS file at the top-level directory of this distribution
 # License: GNU General Public License version 3, or any later version
 # See top-level LICENSE file for more information
@@ -96,13 +96,9 @@
 
 def log_replay_retry(retry_state, sleep=None, last_result=None):
     """Log a retry of the content replayer"""
-    try:
-        exc = retry_state.outcome.exception()
-        attempt_number = retry_state.attempt_number
-    except AttributeError:
-        # tenacity < 5.0
-        exc = last_result.exception()
-        attempt_number = retry_state.statistics["attempt_number"]
+
+    exc = retry_state.outcome.exception()
+    attempt_number = retry_state.attempt_number
 
     logger.debug(
         "Retry operation %(operation)s on %(obj_id)s: %(exc)s",
@@ -117,11 +113,8 @@
 
 def log_replay_error(retry_state):
     """Log a replay error to sentry"""
-    try:
-        exc = retry_state.outcome.exception()
-    except AttributeError:
-        # tenacity < 5.0
-        exc = retry_state.exception()
+
+    exc = retry_state.outcome.exception()
 
     with push_scope() as scope:
         scope.set_tag("operation", exc.operation)