Discovering a task supervisor local to my supervision tree

I was writing a GenServer and quickly discovered the need to generate tasks within this GenServer. This GenServer can not have a globally unique name (as I require many instances of it. I decided to keep the current module structure, changing the GenServer to a Supervisor and spawning my original GenServer as a child of this supervisor (with the Supervisor having public functions that call the child GenServer). Ideally, my supervision tree would look something like this

     MyServerSupervisor 
              /\
             /  \
            /    \
   MyGenServer   MyTaskSupervisor

Also, ideally, I would spawn my tasks within MyTaskSupervisor. However, the obvious question arises: how does MyGenServer discover this task supervisor? In other projects, I’ve given this task supervisor a globally unique name, but I can’t do that here.

Looking into the problem, I’ve seen many suggestions to use a Registry, but those require globally unique names. In order to have a registry for my task supervisors, I’d have to spawn the Registry in my root supervisor to ensure all of my processes can find it. It would be a lot nicer if I could somehow register the task supervisor in the context MyServerSupervisor. Is there a way to achieve what I want, or is the closest analog the solution I just described?

Thanks!