I wonder if it's not the jack, but perhaps something else. Maybe others can chime in and say what it might be...
But let me make one point clear: You shouldn't need a fan to make the player boot into the correct mode. Temperature shifts might make an intermittent failure easier or harder to reproduce, but that's just a diagnostic, that's not a repair strategy. You have to remember that it is still a *failure* and its root cause needs to be narrowed down and repaired.
If the player isn't booting into the correct mode, then either the PIC that controls poweron behavior is getting the wrong signal, or Hijack is being confused by something and it thinks it's being told to change the player's mode into AC mode after bootup.
If the wrong signal is going to the PIC, the most common source of that is a bad AC adapter jack. But I'm wondering if there are other ways this might go bad. Those who know the motherboard circuitry: What happens to that control line after it leaves the AC adapter jack? Does it go straight to the PIC or some other IC? Is it possible that a dry solder joint on one of the IC pins could cause intermittent AC-power-detection? If so, which one should we look at?