The save issue is really giving me a headache. I'm seriously considering removing the Broswer version. (But that's also not an option. A lot of people already played the game in the browser.)
I've temporarily added a note in the guide saying "exe version is safer for saving," but honestly even I would probably skip reading it.
I tried to reproduce the issue you mentioned but couldn't get it to happen. I need to reproduce it to even find clues to fix it. Since it saves to each browser's localStorage, the data shouldn't disappear unless you clear cookies/cache. At least that's what I know.
I tried closing/reopening the browser and refreshing, but no luck. I'll dig a bit more. Really sorry for the save problems.
Testnig with Chrome, Edge, Firefox, Brave, and Epic Privacy Browsers. All of them are ok in my environment..