Answer
Yes. Open a conversation's More menu and choose Mark as Read or Mark as Unread. Mark as Read clears the unread dot up to the latest customer or HelperHat message. Mark as Unread brings the dot back on a conversation your team already read, so you can come back to it later. To mark several conversations as read at once, select them first.
Read and unread are shared by your whole team. Everyone sees the same unread dot, and an unread conversation in Inbox shows the red bell on the Inbox tab.
If you mark the conversation you have open as unread, it stays unread while it's open. It becomes read again when a new message arrives, when you reply, or when you open it again later. If the change can't be saved, the conversation goes back to how it was and a message lets you know.