Swup.js and Snipcart - How to reload "snipcart-items-count" & "snipcart-total-price"

Hi, there!
This is my first question here.

I am trying to integrate Snipcart into a website that uses the Swup.js page transition library. Everything works fine (with some compatibility fixes) except Cart summary which shows data only on the first page load.

I saw there are similar issues with React integration, but I just don’t succeed in creating the solution.

Any help or hint is highly appreciated.

Thank you in advance.