Intermittent AC issue

[Clément161206]

Hi everyone,
I’m turning to you because I’m experiencing a rather annoying AC issue on my Alfa Romeo Giulietta III (2011 model - 1.4 T-JET 120hp petrol - manual gearbox).
The issue:
It’s "all or nothing". Either the AC works perfectly for the entire trip, or it blows no cold air at all, just ambient/warm air.
Behavior:
If it works at startup: It will work flawlessly for the entire journey without any interruptions, even if I drive for hours.
After a short stop (around 5 mins): If I turn off the ignition to run a quick errand and start the car again, the AC kicks back in and still works fine.
After a long stop (several hours): It’s completely random. On the next startup, it might decide not to blow any cold air at all for the rest of the trip.
What has already been checked:
Refrigerant level: A full recharge was done last year to try and fix this, but the abnormal behavior remained exactly the same.
Interior ventilation: The blower fan works perfectly. When the AC fails, air keeps blowing normally (just warm).
External temperature sensor: The temperature displayed on the dashboard is always accurate and matches reality, even when the AC glitches.
Engine temperature: The failure never happens while driving due to engine heat (no cutting out after 15-20 mins of driving). It really comes down to the initial startup phase.
Given the symptoms (losing its calibration/settings after the car goes into deep sleep, but working fine once it successfully starts), I suspect either a failing relay or the climate control blend door gears slipping during the startup self-calibration phase.
What do you think? Has anyone dealt with this exact issue before?
Thanks in advance for your help!