Recently my Little Dot MK III SE has developed a short in the headphone jack. It can re replicated with different headphones. I know the jack is soldered on the pcb. I have the tools at work to remove and replace the jack.
You need to replace to Soviet tubes in the back. I had the same issue with mine, and I screwed around with it for days. Eventually, as a last ditch effort, I replaced all of the tubes (replaced the back with some NOS....stock tubes, and the fronts with some GE EF92s). The problem stopped. Also, EF92 <3.
For what it's worth, I discredited that idea at first because I heard it in both ears (being balanced tubes...you probably know why), but either there's some sort of electrical thing going on between them in the chain, or else being paired NOS, they, against odds, did the same thing at the same time.