Tailor blueprints multisite categories problem

Hello! I’ve encountered some strange behavior with tailor blueprints in multisite mode. There’s a post that translated fine into all languages. I create a category in the main version of the site, then switch to the versions in other languages and add categories but in their respective languages. The categories all show up fine. Then when I try to assign already created publications to categories, something strange happens. I assign a publication in Russian to a Russian category, save it. Then I go to the English site, assign the publication to an English category, save it. When I go back to the Russian version, the category is not assigned.

I am using the default blog blueprint scheme that comes with October CMS after installation.

Maybe my answer to my own question will help someone, how I solved this problem, I’ll post screenshots