Yes, this new set of password worked.
On that site, WordPress and others (Plugins, Theme) are outdated, I'd suggest to update them.
On the Form you mention, I saw that you use AJAX submit (I did not use that).
However, I tested the form on a new Page and I can submit it just fine when the field is hidden.
I see you make that field not required, so there is no reason it should stop the form from submit and fail as required, as the description here says:
https://toolset.com/forums/topic/field-not-unhooking-when-using-conditional-html-output/#post-1113077
Of course, if you hide the field, the field has no value and won't update anything, as it's empty.
If you instead show it and populate it, it will be saved.
However as in the BUG, I reported to the Developers, validation will not apply currently to the Fields.
Means, the form will submit even if the field is required but empty.
During my report, I found that this does not only happen for Post Reference Fields but for all kind of Fields validated by Types as required.
If you encounter other issues, I suggest opening another ticket.
The problem that the field cannot be submitted if hidden, I could not replicate.
If I did the wrong steps, can you correct me where needed so I can try again?
I did insert the Form here:
hidden link