I just had ten minutes spare to check this. I put a voltmeter on the white wire that you mentioned and ignition off it's 0V and ignition on it's 12V, irrespective of where the gear lever is. So I have messed up something.For the starter inhibit in gears other than Park and Neutral, the V6 uses a white wire from the auto box to the Immob ECU.
Now, the Immob ECU prevents cranking when the ignition key chip is incorrect too and also requires a key chip match to the Immob ECU and the Engine ECU.
Have you retained the key chip and Immob ECU as originally designed or bypassed any of it?
Just wondering if any mods have eliminated the Immob ECUs ability to prevent cranking. It does this by breaking the feed to the starter relay.
I guess the white wire provides a feed or 0V only when in Neutral or Park, If this is true then an additional relay in the starter relay path controlled by this wire would solve the problem.
The immo and key chip is unmodified, present, and it's the original K series one.