This happens because ZenDesk's settings for chat routing are way too convoluted ever since they merged chat with the agent workspace.

There are ways to set this up, and I had to spend days figuring this out, so I'll try to save you the trouble:

1. You need to setup in chat routing, that chats are routed to ALL

2. Then you need to set up chat departments (this is important) in ZenDesk chat (one department for each support group)

3. Lastly you need to setup support triggers to route chat Tickets to specific groups.

This one is more difficult as you probably will have to identify which Ticket belongs to which group.

With this setup, when a chat request comes in, only the agents in their respective group/department that the Ticket was assigned to will receive a notification in the top right corner. (The agents HAVE to set their status to "Online" in order to receive notifications.)

I hope I was able to help.