Thanks for your swift response.
I trigger myself the conflict resolution thanks to two separate accounts (since it is my dev wiki) on two separate browsers (I tried a bunch, does not seem to be the cause of it).
I run MediaWiki 1.35.6 on a local EasyPHP dev server (Apache 2.4.25 x86 - PHP 7.4.19 x86 - MySQL 5.7.17 x86).
Javascript is enabled and works fine. No particular message in browser console either.
The error happens whether there is more than one paragraph or just one.
It seems that if I choose the newest, not saved yet, revision, the page reloads with a red "alert" background-color on conflicting paragraphs and deletes all choices.
If I choose the already saved revision, the paragraphs that had conflicts just disappear, but the page is still not saved: the interface shows then only the unchanged paragraphs, which is really weird.
I hear you about the PHP version, but every other extension works perfectly, so that would be odd.
Might be linked to my other extension configurations, I'll try on a fresh install and keep you posted.