If any installed app has push messaging permission, then background launch Chrome

chrome/browser/background/background_application_list_model.cc

Index: chrome/browser/background/background_application_list_model.cc
diff --git a/chrome/browser/background/background_application_list_model.cc b/chrome/browser/background/background_application_list_model.cc
index 08b6d1c6d7f94141ae6368aa7ba932e102e6f999..e0c6010b6ff1f608cc3f3bcf85080be4aef3d8d4 100644
--- a/chrome/browser/background/background_application_list_model.cc
+++ b/chrome/browser/background/background_application_list_model.cc
@@ -277,8 +277,10 @@ bool BackgroundApplicationListModel::IsBackgroundApp(
   // 2) It is a hosted app, and has a background contents registered or in the
   //    manifest.
 
-  // Not a background app if we don't have the background permission.
-  if (!extension.HasAPIPermission(APIPermission::kBackground))
+  // Not a background app if we don't have the background permission or
+  // the push messaging permission
+  if (!extension.HasAPIPermission(APIPermission::kBackground) &&
+      !extension.HasAPIPermission(APIPermission::kPushMessaging) )
     return false;
 
   // Extensions and packaged apps with background permission are always treated