Unfortunately, the fact that the core WooCommerce Cart widget exhibits the same behavior tells us that the issue is not with WP Menu Cart…
Both me and @yordansoares did several tests on your site and the behavior was quite inconsistent accross the board. Overall, it seems that the first time you open the page in a new tab (clean session), the WooCommerce AJAX system (“fragments”) does not work – it is stuck on the empty cart and is never refreshed. When we then opened the site in a new tab in the same session (after having added something to the cart in the original tab), everything worked perfectly: WooCommerce AJAX fragments were updated everytime something changed in the cart.
We were not able to find the root cause of this. There could be a server side issue with sessions, or an issue with cookies on the client side, but what exactly is unclear and like I said: the fact that this happens with the cart widget demonstrates it’s unrelated to WP Menu Cart.
My advice would be to test 2 things:
- Try with all plugins except WooCommerce disabled (you could enabled WP Menu Cart too if you like), make sure to start your test in a clean session (incognito)
- Try in a different server (not your own) to see if it’s perhaps server related
Hope that helps – let us know your findings!