I've always felt both were correct but I've also always seen people on mumsnet say only "arsed" is correct.
Those who say "can't be asked" makes no sense are simply living with what they've known. "Can't be asked" makes much more sense than 'can't be arsed', which comes across as more regional than 'asked' in terms of rational meaning to the outsider (someone who's never heard it or not used to it).
"Can't be asked" - Of course you can literally ask but I won't do it. Can't be asked to do it because I can't no matter how many times you ask.
"Can't be arsed" - Qué?
"Can't be bothered to do it" - Makes sense. Same as can't be asked. You can bother/ask me about it but I won't/can't do it.
Only when you're used to arsed/bothered as interchangeable do you allow it to not matter in your brain. Besides that, "arsed" makes no sense at all.
It's a bit different from "I'm not bothered" (to do it/arsed to do it). 'Arsed' still sounds ridiculous or regional but fine as substitute.
"I'm not asked" (to do it) then finally makes no sense.
This is all to an untrained eye who's not learned that "can't be arsed" is normal their whole life. So there's some overthinking going on about it.