In my company, we are encouraged to submit ASRs if we have a callsign conflict with another aircraft, whether it's with one of our own company or another one. I'm told, although I don't know how it works, that the process of getting it changed is relatively simple once the right people have been made aware.
Do ATC units not have a similar facility, to allow them to report conflicts to the relevant people (airline flight planning departments or whoever) and get the callsigns changed accordingly?