We have experienced a similar sitauation. We get a lot of leads emailed to us from voice-noreply@google.com and other no-reply type addresses. These lead emails accumulate together under the relevant noreply Contact. TheContact Owner of this record is supposed to remain “unassigned”. Whenever an employee accidentally does a “reply” to one of these emails via a connected inbox, HS assigns that that employee as the Contact Owner and all subsequent leads from that “noreply” email address bypass our Conversations Inbox and go straight to that person. This was very inconvenient and caused fresh leads to be overlooked because they get assigned to someone in Customer Support instead of Sales.
Here is how we partially solved our particular problem. You mileage may vary:
Create a workflow like below:
Now any Contacts with “noreply” email addresses will never have a specific Contact Owner - at least not for very long. Workflows sometimes take minutes to execute, so there is still a window for errors to happen. But at least that window is now minutes and not hours or even days.
This solution might not be optimal for you.