Cookies or LocalStorage for Language selection using subdomains

Hello community! I am trying to make a selection of the country on the site and then switch to another subdomain. I used to save the selection to Local Storage, as in this tutorial from Finsweet: https://www.finsweet.com/hacks/37/
Works well, but I ran into it not working between subdomains. Tell me how you can make the same choice appear after switching to a subdomain?

Here is the link to page: https://langselector.webflow.io/

And here is the read-only link: https://preview.webflow.com/preview/langselector?utm_medium=preview_link&utm_source=designer&utm_content=langselector&preview=4e86f0912f7b0af190cf26b1080a9d30&workflow=preview