Sporadic issues accessing 'https://<space-subdomain>.hf.space' URL on a Private space

After a user Duplicate one of my Public Space (OMNITOOL.ai on Hugging Face - a Hugging Face Space by omnitool-ai, to be exact ) and create the Duplicate as Private, they sometimes get a 404 error when accessing that ‘direct’ URL (e.g. of the type: https://[space subdomain].hf.space )

I’ve confirmed by creating a test user that this sometimes occur - it seems, randomly.
In my case, after toggling the space from Private to Public and back to Private, doing a Factory Reset, etc., the problem went away for me but not for some users. On most of my tests though, both with existing accounts and brand new test accounts, the problem does not occur at all and the direct URL of the Private Space is accessible (with the user being logged in and owning the space, obviously).

My question is: Is accessing a Private Space through the ‘https://[space subdomain].hf.space’ URL by a logged-in Space’s owner something that is supported and supposed to always work?
(I’m wondering if this is the case as Private Spacse do not have the ‘Embed This Space’ menu option, which would tend to indicate this isn’t a really supported use-case.)

If that helps, the documentation mentions:

A Space is assigned a unique URL you can use to share your Space or embed it in a website. This URL is of the form: "https://[space-subdomain].hf.space". For instance, the Space NimaBoscarino/hotdog-gradio has the corresponding URL of "https://nimaboscarino-hotdog-gradio.hf.space". The subdomain is unique and only changes if you move or rename your Space. Your space is always served from the root of this subdomain.

Any clarification would be appreciated!