I think the issue could be related to the translation preferences of the theme's custom fields which appears in your page editor.
Please try the following steps and let me know if that helps:
1. Go to WPML > Settings > Custom Fields Translation
2. Set the following fields to "Copy": rz_header_layout, rz_header_primary_menu
3. Save the changes
4. Edit the homepage in the default language and make a small change then save it
5. Update the translation
I would need to request temporary access (WP-Admin) so I can investigate this issue further.
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It seems that the steps I shared are working, I just resaved the homepage translation then the issue was fixed. Now I can see the correct menu when switching to English hidden link