A bit of an update…
I did not do exactly what doncorleoni said, but something slightly similar: I released most of the air from the fork (so it reached about 50% sag under its own weight), then I put the bike upside down and pumped the fork few times. I could hear the oil and air getting mental inside, then I put the bike up, inflated the fork, pumped it down few times and… The problem seem to be gone!
However now when the fork is locked (with me sitting on it) the uppers slide into the lowers about 4 mm (previously there was no movement at all), but from there it is a “concrete lockout”. And no “loose headset” sensation. I’ll test it off road tonight! But this is brazzers… I mean bizarre!!!