[Решено] Compatibility problem with ACF when translating menu

Background of the issue:
When I translate my menu without ACF (Advanced Custom Field), everything is fine. When I activate the plugin, a lot of weird behaviors arise (see my previous ticket on that matter :https://wpml.org/forums/topic/impossible-to-translate-menu-correctly/). Mainly, the menu in the BO disappear from the original version and the translated menu are duplicated (but the display on the website is a weird mix of original and translated entries).

We were finally able to create a staging website so I can give you access to have a look.

Symptoms:

Questions: