When you edit the content of your Persistent menu, sometimes those changes aren’t shown in your page’s inbox right away. This is due to the fact that the Facebook cache gets refreshed only periodically.
What you can do to speed up the process is disconnect the page from your bot and then connect it again.
Before testing again, first delete the conversation with the bot from your personal inbox. That will enable your local cache to get refreshed as well.
If you’ve attempted these solutions, and your Persistent menu doesn’t seem updated - reach out to support and we'll look into the issue.