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That was activated immediately, without that we couldn't login at all.

so, resolved?

No. Yesterday the same thing happened. Suddenly logged out and when I checked cookies, I saw that phpbb3_qac2sy_k is empty. So frustrating...
I cannot figure out the logic behind it. I'm starting to think that it has something to do with phpbb itself, and not with w3 necessarily...

ok ... yes above you have reason isn't easy to understand where the problem can come out due to actions and filters added by several plugins.
More, thousand of options ... but all should return a good result if all are doing theirs work ok.
I do not know what plugins you've install, and which of those cause the problem.

If logging in, in phpBB or WP, you see that the k cookie is correctly released, but after suddenly it is reset and user logged out, you can be sure that if this happen when you visit phpBB it is a phpBB problem (which i do not think in this case, because i assume you have correctly setup in phpBB all setting about security and time session duration).

While if it happen on WP side, you can be sure it happen due to some plugin in WP.
Have you try to empty the plugin cache in WP? (i assume yes)

Please activate my account (no confirmation received) i'll check a thing to see what happen ...

You're activated now.

Sure, I flushed cache often, still the same problem... I'll try maybe without cache plugin at all, for several hours, our VPS should hold on, and that way I'll know for sure if that's the source of the problem.

result? ... i'm in little delay, but over now ... really hot summer here ...