Version Sync needs the Administer Projects permission to work properly. If not provided, the app shows a warning message at the top of the settings page.
When the app fails to sync a version because of insufficient permissions, a log message will be added to the Activity Log.
You can grant the Administer Projects permission by navigating to the project’s permission page and clicking on Actions > Edit Permissions > Grant permission.
Then you choose the Administer Projects permission and select Project Role and atlassian-addons-project-access as shown in the picture below.
After that, click on Grant. Version Sync should now function properly.