Message about previous submissions for anonymous users

Hi, I'm trying to allow anonymous users who had already submitted a specific webform to see their previous submissions, including the message about existing previous submissions, similarly to authenticated users.

After allowing premission to view (own) submissions, this seems to behave a bit randomly, and I'm wondering how previous submissions are determined for anonymous users. When submitting anonymously from same device and location, sometimes the message about previous submissions is displayed (attached to the confirmation) and sometimes not. If the page is reloaded (or left and then entered again), the message is never displayed, unlike authenticated users for which the message is displayed whenever the page is entered.

I'm not entirely sure whether this is a bug, a support request or a feature request... any help would be most appreciated. Thanks!