Notifications customer dialouge

i am searching for a functionality which helps me to have a view over the customer-dialouge.

We have set up only one e-mail address - and below a task or an issue i can send a new email to the customer. That works fine - if the customer answers, this mail is now to find also below the infos inside the task or issue.

But, how anybody in the team can recognize, that the customer sends an answer?
OK, in issue, it is working, because the status changed and after an answer it is again status open.

But how does this work in tasks? There are too many status open, if we set up e.g. a new project … nobody knows how many tasks are open there, to regognize, there is an answer inside one of them.

In this case i thought, there is the icon if the email-inbox. but the new answer inside a task will not appear there.
Sure, i can set up an e-mail alert - but it is not the goal, to need a standard e-mail account to recognize that there is an answer inside a task.
is there another solution, like as the e-mail inbox, where all task-mails are notificated ant the user has to change the status to close and thats it?
thanks for your ideas