Writing directly to a side conversation

Hi, one of our customers uses side conversations to communicate with us (IT serviceprovider).

Unfortunately, our own ticket system (Autotask) can't simply answer those emails, but is instead creating a new email, which is removing the header information.

Therefore, an answer is always generating a new ticket in ZenDesk on our customer side. To combat this, we tried to write to support+id<ID>@customer.zendesk.com. This results in the mail getting to the correct ticket, but not in the side conversation. Is there a similar address to write directly to the side conversation (like i.e. support+side_id<side_id>@customer.zendesk.com) or any other solution that doesen't involve using either the header or involves something like parsing the header information on our side and use some of it's information in the subject or body of the mail? Thank you in advance and best regards, Marcel