My guess is that an in-tank pump may be fed through a connector that could also take in the fuel gauge, perhaps both using the same earth (black). That perhaps being the case then green/black may be the fuel gauge thus looking to find the white/purple as being live with the IGN switched on would seem a worthwhile check..? Assuming there are no other safety devices/switches between the ign switch and the inertia switch you mention.
In my car the oil pressure has to be present for the pump to continue running and on fuel injected cars there can be other measures used to ensure the engine is running to ensure that the fuel pump electrical supply is maintained.
Good luck.
