Thanks for your help guys, it's much appreciated!
Back to it, I don't see any bad solder joints (and I also tried to resolder the headphone jack with no change)
The headphone jack is soldered into a very small PCB (which has two resistors, R1331 and R1332, and two special, resistor-shaped capacitors, named C1331 & C1332) and is then connected to a larger circuit board via a 3-conductor flat cable (FW251), which is soldered on both ends (tried to resolder that one too, but again, didn't make a difference). Apart from that, both R1331 & R1332 are fine and shows 470 OHM on my meter.
C1331 & C1332 seem to be okay also (desoldered and checked capacitance, both were on the same level).
FW341 (in similar to FW251) is also a flat multi-conductor cable (except that it has 5 conductors instead of 3) which connects it to another circuit board which does all the processing and has the two main, big capacitors. Haven't tried resoldering that one yet, maybe will do that too soon.
There's also a transistor (A1046) on the beginning of that 3rd circuit board (right next to the FW341 cable), I'll have it tested later, maybe it's somehow related. As for the relay (RY203) I'm not sure how to test it (as I haven't tested any before) so some guidance would be great!
Apart from that, I only have a multimeter (UT139C), a capacitance+esr meter (ESR70) and a soldering station (FX888), so no scope, please keep this in mind. Also, not sure if it matters, but I've noticed that if I turn the volume really high, there is very (slightly audible) output from the left side, so it's not completely muted.