Just called and got right through (on the no-status line). Agent refunded ticket without question. Got the standard 7-10 business day spiel.
Good news! I received a refund today for one of the tickets in our two-passenger booking (Italy round-trip in May). I expect the other ticket's refund to post shortly.
EDIT 5/16/202: The other ticket refund posted to my credit card last night.
RNE, reporting that it took 23 business days from the date UA agreed to the refund.
P.S. I'm still waiting for the return of seat upgrade fees to the tune of $600.
P.P.S. Keep fighting!