Highly unlikely a pilot would lie about a radio issue.

I will say that at CLT, I was fortunate to have a late, last flight to ORD held for us and 10 +/- others, when our incoming flight was delayed 30 minutes due to a mechanical issue. The pilot made up some timeso the flight to ORD was only a few minutes late to the gate. It was a obvious benefit to AA by holding that flight that probably saved them a few $$ in hotel and flight rescheduling issues, plus generated a bit of good will for many.