I discovered another problem with the forum.
If I activate the function: "Turn WordPress to wpForo", everything works normally.
But if this function is not active, I have a problem with the cache.
The user logs in and everything seems to be ok. However, if, after logging in, he goes to the homepage, for example, then everything looks as if he has been logged out.
Or the logged in user cannot log out.
Some pages are logged as logged, others logged out. There is confusion in that.

Everything on the web of the current version. WP 5.9.
There is an oxygen builder on the web who has a CSS cache, but I tried to turn it off and I also filled out oxygen. The problem remains.
There is WP Rocket on the web, but it has the /community/ (.*) check box turned off. I tried to turn it off completely, but the problem still persists.

Other suspicious plugins on the site are not used. But still, I tried to deactivate them all and the problem persists.

The site was cloudflare, but it is already completely deactivated.

I thought it would be some kind of server-side cache, but hosting says no.
It confuses me that the mode: "Wordpres to wpForo" is all ok.

Thanks for the help.