That loose plug with the purple wire under the infill panel on rear left of engine is for an optional accessory, the 12v socket iirc. Not related to GPS anyway.
Your problem given it's a 2011 Roadster does indeed sound like the good old GPS fault with 2010 - March 2013 Roadsters.
The GPS is located near where your left heel is when riding. Its cable will have 3 wires I expect. Held on by 2 bolts. It's not covered by anything. You only have to remove that infill cover (which is above the GPS) when installing the new GPS, to access the plug.
That thread (if read) says you have to replace the GPS (new one has 4 wires, plus a 'fly lead' / adapter harness for 2010 - March 2013 Roadsters), and then make that wiring adjustment to the tachometer harness plug.
It also notes to be sure the fly lead is plugged in between the primary TPS plugs and not the speed sensor, it is an easy mix up because they are similar plugs and located close together.
The tacho plug wiring adjustment will not correct it on its own. Perhaps that thread needs a few more pictures..