This can certainly happen when a contact is logged in already and their membership levels change. We do cache their membership levels at the beginning of each log in to prevent delays as they load each page. There is a way to force a refresh automatically.
You can force a refresh when they visit the page by inserting the special shortcode [pilotpress_sync_contact] at the top of the page in Text mode (classic editor) or in text mode in a Gutenberg "classic block". We have full instructions on this option in our Shortcodes article in the Resync Contact and Sell Membership Usecase section.
Here's how the drop down for the merge fields appears; the Resync Contact option is the last one in the drop down: