diff --git a/scripts/verifyProxy.py b/scripts/verifyProxy.py
index 7e76b7c..c588802 100644
--- a/scripts/verifyProxy.py
+++ b/scripts/verifyProxy.py
@@ -240,6 +240,12 @@ def main() -> None:
             time.sleep(2.2)
             print("PASS Nginx TLS proxy, Keep-Alive and upstream timeout", flush=True)
 
+            # Drain the edge proxy before stopping its upstream, matching a
+            # rolling deployment where traffic is removed from the old task
+            # before its listener is shut down.
+            nginx.terminate()
+            nginx.wait(timeout=5)
+
             assert backend.stdin is not None
             backend.stdin.write('"stop"\n')
             backend.stdin.flush()