There is actually an another issue that I've been working for is the custom post list お知らせ (news) page in my theme doesn't work properly. The list is not showed in each languages but showed in all languages mixed up.
I attached images of the list and of the admin page.
It will be grateful if you could check out as well.
since you are using a custom-coded theme it might be a bit out of scope to debug and check your custom code, but I will do my best to suggest the best solution or approach you can use to solve this issue.
Can you please elaborate with more details on how are you fetching and showing this part with posts? Using some custom code?
For this issue, I would like you to check out my site but I have another issue to solve by other supporter at the moment. It will take a time but please wait for a while.