diff --git a/.github/workflows/downstream-artifacts.yml b/.github/workflows/downstream-artifacts.yml
index 48e958bec49..b16528cb66c 100644
--- a/.github/workflows/downstream-artifacts.yml
+++ b/.github/workflows/downstream-artifacts.yml
@@ -3,8 +3,16 @@ on:
     pull_request: {}
     merge_group:
         types: [checks_requested]
-    push:
-        branches: [develop, master]
+
+    # For now at least, we don't run this or the cypress-tests against pushes
+    # to develop or master.
+    #
+    # Note that if we later choose to do so, we'll need to find a way to stop
+    # the results in Cypress Cloud from clobbering those from the 'develop'
+    # branch of matrix-react-sdk.
+    #
+    #push:
+    #    branches: [develop, master]
 concurrency:
     group: ${{ github.workflow }}-${{ github.ref }}
     cancel-in-progress: true