How do I prevent a public comment notification if an agent is replying to the thread from outside zendesk (Outlook/Gmail/etc)

In our organization, sometimes the CC'ed users (mostlly our internal team members who are not on zendesk) try to create a private email conversation without the customers. And if a manager with Zendesk license replies to that email, all the conversation is sent to zendesk on that thread.

Now I understand that this is because one of the Zendesk support address is added to the email conversation but is it possible to prevent a public comment notification to go out to the customer if the update channel is outside of Zendesk? I don't see any such distinction in the triggers yet.

I understand that We have Mail APIs but I cannot expect all our team members around the world to remember them to send email.