Auto refresh of a password protected page

In WordPress, is it possible to auto-refresh a password protected page, such that the password only needs to be entered the first time the page is accessed?

I know how to password protect a page.

I know how to auto refresh a page.

Unfortunately if I auto refresh a password protected page it asks for the password every time the refresh occurs.
Is there a way so that the password only needs to be entered the first time the page is accessed, and not on subsequent refreshes?