Who/whom question:
"I have no idea whom he was speaking about."
The subject of the phrase, "was speaking about" is, he, hence "whom" is correct (flip it around, "about whom he was speaking').
But it doesn't sound correct.
Who/whom question:
"I have no idea whom he was speaking about."
The subject of the phrase, "was speaking about" is, he, hence "whom" is correct (flip it around, "about whom he was speaking').
But it doesn't sound correct.
It does sound wrong imho. So why not flip it around as you suggested "about whom he was speaking", or better still, "of whom he was speaking". You get to avoid a trailing preposition hehe...