This is the expected behavior of the conversations tool, and it's designed this way to ensure a more cohesive customer experience.
Each time a rep responds to an unassigned conversation, the email will be assigned to them and will stay with this rep until the conversation is closed. If the customer follows up, the conversation will also route back to the rep that assisted them to ensure that the customer doesn't get bounced back and forth with different reps.
Would you be able to tell us a bit more about your use case, please? Are you looking for a way to leave each of the customer's replies "unassigned"?
I can give you the use case. I have been searching your help for the last hour trying to figure out a way to handle this:
A rep goes on vacation. They had previously closed a conversation with a customer. The customer replies to that conversation. Since the rep is on vacation, it will sit in their queue, unactioned on. We have to manually check every rep's queue or always look at All conversations. We do not use tickets as they are too much work for too little gain.
Additionally, I'll just say that it is frustrating as a user to have this inconsistency between conversations and the rest of the product. This is not the first time I've thought "Oh, I'll do this little bit of automation to our inbox" only to discover that I can do that automation in tickets or deals, but not where I actually do my work. I don't know what Conversations is supposed to be in the eyes of Hubspot Product Managers, but it doesn't seem to match what it is in the community discussions I've read.