I download and run the game itself trough the "Heroic Launcher" as flatpak on my Debian-based distro, and it runs good.
The problem is whenever I save the game nothing happens (at the first phone cell in chapter 1).

When saving the game, it does not respond at all, like I don't get the message that it's done and I have to start the game every time from beginning. I read, that this kind of glitch may occur and I tried it few time with re-installing the game and doing it over and over again, but the game just wont save.

I am not sure, if this post will help you.

I have kind of the same problem in Windows.
Sometimes the game doesn't save, even when I pressed every button correctly.
In this case the message "Save complete" also doesn't appear.

I just try again, and most of the time it works the 2nd time or 3rd time.
But I guess you already tried that.

Good that the main problem for Linux also got solved!


Still, my Windows problem might also apply to you:

When saving, pay attention, that the message "Save complete" appears.

If yes, everything is ok.
I no, you have to save again (and again), until the message appears.

It is slightly annoying but not a big problem when you know about this.