1.9 TDI BLS engine stalls at traffic lights when the engine is warm

Hello, I have a Volkswagen Golf V 1.9 TDI 'Pumpe-Düse' (pump injector) from 2006 with only 24,000 kilometers on it.
When the engine is cold, it starts immediately. The warmer it gets, the harder it becomes to start.
When it reaches almost 90 degrees Celsius, it shuts off and won't restart. You have to wait a long time for it to cool down before it will start again.
When the engine is cold, there is no air in the return line. As it warms up, small air bubbles appear in the return line. The warmer it gets, the more air there is.

It was done.
New seals installed: no success.
'I swapped out the PDE and tandem pump with parts from another car: still experiencing the same problem.'
Fuel pump in the tank, new: didn't help.
Removed the diesel filter and connected it directly, but no success.
Compression measured: OK.
PDE seat ground, no change.
'When driving at low RPM, there's a significant vibration. When the engine is warm, it stalls at low RPM. If you keep the engine above 1500 RPM, it stays running and drives normally.'

Does anyone have any ideas?
I welcome suggestions for solutions!
Thank you for your efforts.

It comes with a diagnostic system that integrates into the engine control unit when it fails.
Errors saved. (Autoscan)
If it dies and you turn off the ignition, the glow plug indicator will briefly light up.

If you read the error codes, there are no errors. All the lights on the dashboard always illuminate normally.
I fitted the supply and return lines with a long, transparent hose. The supply line is always free of air. However, the moment it stops flowing, air enters the return line. When the engine gets colder, the return line automatically refills with diesel, and it starts up normally again.
'When you unplug the EGR valve connector, the engine runs a little better and idles more smoothly.'
The particulate filter is only 20% full.
When the engine is warm and stalls, then fails to restart, and you try to start it again by engaging the starter motor, the fuel pump in the tank runs continuously.
Here's another test: When the engine is cold and I remove the glow plugs and turn the starter, a mixture of air and diesel comes out of all cylinders. However, when the engine is warm, has been turned off, and I then remove the glow plugs and try to start it, only air comes out, meaning that at that moment it's not getting any diesel because the cylinder head passages (visible through the return line) are full of air or diesel vapor.

It sounds like real steam bubbles, but those can completely disappear through condensation. Air cannot do that.
Is the diesel mixed with volatile liquids like gasoline or similar substances?
Has anyone recently put the wrong fuel in their car?
Is there a vacuum in the fuel system? (During troubleshooting, open the gas cap and listen.)
Best regards, gnash.

The diesel and the steam coming from the return line smell like diesel. I recently bought it with this problem, and I'm not sure exactly what fuel was put in it.
I loosened the fuel cap. There's no difference, and I can't see any flap behind the fuel cap; I have no idea why.

Have you carefully examined the PDE in the front area of the density plate (using a microscope or magnifying glass)?

I once had a situation with a Deutz engine where an injector nozzle had a crack (barely visible to the naked eye), through which combustion gases were being blown into the fuel line.
It would only run as long as it was cold.

EDIT: What is the function of the crimp connector (ground) on the cable harness located under the valve cover?

I would first drain the diesel from the tank and then refill it with fresh diesel. That way, you can be 100% certain that there is actually diesel in the tank.

Otherwise, it sounds like a hairline crack is expanding during heating and closing again during cooling. In this case, I would try to gain clarity by performing a pressure decay test.