Alright, I'll check all the capacitors. I've already tested a few and they were fine. Desoldering them and checking their capacitance should be enough, right?
No, completely pointless, it's the ESR which goes high - you need an ESR meter to test them (as augustinetez mentioned).
It's extremely common that their capacitance is still fine (and within spec.), but the ESR is what matters - I've tested (and replaced) thousands of such failed capacitors. Basically it's down to cheap low quality capacitors, if decent quality ones were fitted, then they
DON'T fail - if cheap crap ones were fitted (like Samsung have tended to use) then they fail pretty rapidly.
However, assuming it's the PSU which is at fault, it could be...