"gearbox is in emergency mode" fault on dashboard but disappears quickly

I occasionally see a warning message that "gearbox is in emergency mode" when I switched on the car. It only occurs during ignition and have experienced during the mornings (i.E. When car is cold). It does not happen every day though.

I took it firstly to VW they say a selector lock is missing and a new solenoid needs to be fitted. They did though say that it might not fix the problem. So I was not comfortable paying around 950GBP for something they was not sure about will fix it.

I then took it to an automatic transmission specialist who diagnosed the car. He could see the warning comes from the solenoid, however, he told me nothing is wrong with your gearbox. He believes that is it related to my battery and alternator. He did a battery test which said replace (I think about 50%).
Also, the alternator was not giving full voltage. When he rev engine fully, it was still missing a few voltages ( he said 2V or so).

The conclusion was that when igniting the car in the morning not enough power goes to all units, including the solenoid, making the car believe that something is wrong with it during that fraction of second when the warning is up?

I still feel a bit not sure, and was wondering if somebody does have any experience with this

Just to add:
The error code in the transmission module is:
P092800 GEAR SHIFT SOLENOID CONTROL CIRCUIT OPEN

Below he writes:
LOT'S OF COMS FAULTS AND UNDER VOLTAGE CODES.