This thread is resolved. Here is a description of the problem and solution.
Problem: The client is experiencing an issue where pages that take data from MetaBox display the correct translation in Greek, but when switching to the English version, the Greek translation persists. This problem only occurs for logged-out users; logged-in admins see the correct translations in both languages.
I migrated the site to hidden link and tested some workaround without any luck. So I've escalated the matter again to our 2nd tier team for further investigation.
which only registers the CPT (without the extra logic to flush permalinks and stuff). You can find this code in your staging site -> Code Snippets.
Since Oxygen disables current theme logic, this is the only way to execute some code or we could add this to theme's functions.php file (or write a new plugin).
After that they did some manual translation by editing the template in English (used on hidden link) and you can see it displays that translated (see attached).
Also perhaps it is not a good idea to translate the content with native editor (oxygen builder).
They further advise to contact Oxygen support for proper compatibility. Since this plugin isn't compatible with WPML (https://wpml.org/plugin/oxygen/), you can provide these details to them.
Theme and plugin authors are welcome to join WPML’s Go Global program, which helps them make their code multilingual ready. Together with WPML’s compatibility team, they can test and confirm that their theme or plugin works with WPML. See https://wpml.org/documentation/support/go-global-program/ for details and you can refer your theme/plugin provider to this as well.