2001 S8 gearbox limp mode only on cold start

[jimh]

I have a problem with my 2001 S8 auto transmission.
Only occasionally, and only on cold start-up, the gearbox goes into limp mode.

I will turn on the ignition, and the P will light on the dash as normal. Start the engine, and it goes immediately into limp mode and lights all the gear lights on the dash. Switch off the engine and immediately re-start, and all is ok again.
The gearbox works fine at all other times.
When in limp mode I only have 4th gear and reverse, and selecting either is harsher than normal.

I have checked & cleaned the plug for the range switch on the gearbox, and checked & cleaned the plug at the TCU.
This same problem occurred last winter, was fine all summer, now the weather is cold, its back.

Gearbox fluid & filter was changed about 20K ago. Car has approx 150K miles now.
I'm thinking it could be the gearbox switch, but I'm also suspicious of the battery. Anyone had a weak battery cause this?