I have this same condition in my Mark I.
The guys sent me a new power jack and I installed it but I still have the same problem (always thinking it's at home).
I know the connections are good because I'm getting good continuity throughout.
For some reason the unit also does not want to power up while on AC unless the outside of the barrell connector ( negative side) is touching the edge of the case.
I also seem to get a low power condition quite a bit while it's docked in the car.

Will a blown buffer chip or broken circuit trace cause this condition in mine?

Steve