diff --git a/distributed/active_memory_manager.py b/distributed/active_memory_manager.py
index 790164385c4..4ed2daf4113 100644
--- a/distributed/active_memory_manager.py
+++ b/distributed/active_memory_manager.py
@@ -291,9 +291,18 @@ def run(self):
                 continue
 
             desired_replicas = 1  # TODO have a marker on TaskState
-            ndrop = len(ts.who_has) - max(desired_replicas, len(ts.waiters))
+
+            # If a dependent task has not been assigned to a worker yet, err on the side
+            # of caution and preserve an additional replica for it.
+            # However, if two dependent tasks have been already assigned to the same
+            # worker, don't double count them.
+            nwaiters = len({waiter.processing_on or waiter for waiter in ts.waiters})
+
+            ndrop = len(ts.who_has) - max(desired_replicas, nwaiters)
             if ts in self.manager.pending:
                 pending_repl, pending_drop = self.manager.pending[ts]
                 ndrop += len(pending_repl) - len(pending_drop)
+
+            # ndrop could be negative, which for range() is the same as 0.
             for _ in range(ndrop):
                 yield "drop", ts, None