I know that the only officially supported distributions are Ubuntu 14.04 and 16.04, but I was able to play this game just fine on Fedora 23. However, after upgrading to Fedora 24 (and now 25), the game first fails to start with:
$ ./start.sh
Running Agatha Christie - The ABC Murders
./The ABC Murders.x86: symbol lookup error: /lib/libxcb-dri3.so.0: undefined symbol: xcb_send_request_with_fds

After moving away lib/libxcb.so.1{,.1.0}, it dumps core:
$ ./start.sh
Running Agatha Christie - The ABC Murders
Set current directory to /home/rathann/GOG Games/Agatha Christie The ABC Murders/game
Found path: /home/rathann/GOG Games/Agatha Christie The ABC Murders/game/The ABC Murders.x86
Mono path[0] = '/home/rathann/GOG Games/Agatha Christie The ABC Murders/game/The ABC Murders_Data/Managed'
Mono path[1] = '/home/rathann/GOG Games/Agatha Christie The ABC Murders/game/The ABC Murders_Data/Mono'
Mono config path = '/home/rathann/GOG Games/Agatha Christie The ABC Murders/game/The ABC Murders_Data/Mono/etc'
displaymanager : xrandr version warning. 1.4
client has 4 screens
displaymanager screen (0)(HDMI1): 1920 x 1200
Using libudev for joystick management


Importing game controller configs
support/gog_com.shlib: line 94: 23299 Aborted (core dumped) ./"${bin_64}"

I don't want to revert back to Fedora 23 because it'll be EOL next month. Now, I'm not expecting the developers to rush and fix this immediately, but I'd appreciate it if this was fixed at some point. I'd love to keep playing this game on my favourite Linux distribution. I can provide a full backtrace from gdb if anyone is interested.

Best regards,
Dominik

I have no idea about the game on Linux, but you could try the Windows version with Wine.

Same here on Arch Linux, game won't launch at all.

UPDATE: Apparently it was caused by lack of some libraries, however I'm unable to say which ones precisely. What I did on Arch was installing "steam-native-runtime" package which adds a lot of 32 bit libraries to the system, and after that game started working.

Thanks for the hint. I rebuilt my desktop after a long break from gaming and made another attempt at running this game on Fedora 31. It turns out I was missing the mesa-dri-drivers.i686 package. The game works again, after all!