Marcus - that's an interesting theory about the weights returning the shifter to center. I believe, however, that the function of those weights is to provide additional inertia when shifting gears, to help the synchros work. (It is remarkable how different the 6-speed linkage is from the 5-speed!)
OP - good to know which transmission you're dealing with. Found a related thread here:
986S 6 speed trans issue
Edit: also, on the 6-speed transmission, G86.20, I think the attached picture will shed some light on what's going onside inside the transmission just behind those two selector input shafts.
The spring below the cover marked "A", which I believe also would be part number 996 303 255 00, is what I believe in this transmission would cause the left-right motion at the gear shift knob to return back to center track (3rd-4th gear track). The good news is that the workshop manual does have a procedure for accessing this spring, without requiring transmission disassembly. The bad news is the same spring also causes return from the other tracks (R, 1-2, 5-6) to center (3-4), so I doubt there could be much guarantee that simply replacing this spring will solve your problem. Have you changed the transmission fluid recently?