Most likely you have wiring issue in the right hinge of the trunk. Try pressing the brake while opening the trunk and the glass hatch too or just start inspecting the wiring

 

Like a pinched wire?

 

Yes… it’s a common failure point…

 

Thanks. I'll check it out right now

 

If You find out it is not the cables, and the other brake-, back- light, and indicators work in the back of the car, then You can look at the output of the 3rd (high) brake light, at the J393 electronic unit. If it has any power out on the pin corresponding to the cable.
If not, then Inside it has a VN450 driver circuit, that may have died on that output.

I had One brake light not working.
This was the finding.
https://www.clubtouareg.com/threads/strange-brake-light-problem.300394/?post_id=2169891#post-2169891

or,
maybe a grounding problem?

 

A mechanic friend tells me that the lights must be recoded using a scanner tool anytime a light bulb is replaced.

I would never let this “mechanic” near anything I own.

 

So do you, as I do, feel that he's not even close to being correct?

 

Correct. There is no “coding” of lights necessary. Only if you replaced the front headlight assemblies would you need to recalibrate the AFS.