Very simply I was having the exact same issue you described but with Klei's Oxygen Not Included. Every other game I played the audio was working correctly. Everything else with Oxygen Not Included was working.
First I did everything you mentioned- verifying game files- uninstalling then reinstalling- and nothing changed. I tried reinstalling/updating drivers- nothing changed. I thought maybe it was something to do with the audio jack- unplugged headphones and tried the laptop's speakers- no change. I then noticed that Sonic Studio was controlling my audio like an interface (It had various audio profiles you could choose for each program). I tried changing the profile, no change. I deleted Sonic Studio, and suddenly the audio worked.
Something with Sonic Studio clearly wasn't jiving with how the program was pushing audio. I didn't look into it any further because my issue was fixed.
Have you played Solasta with the volume working before?
I hate to be that guy- but I'm assuming you've checked that your drivers are up to date?
If so, is windows picking up the audio levels from Solasta in the mixer? (the green bars in the picture below)
If windows IS picking up the audio levels, I would suggest maybe trying a different audio output? Checking that windows has the correct audio output/hardware is selected?
I would suggest copying your saves, then doing a clean install of Solasta as well.
If none of that works- then. Well. I don't know. Could be a weird quirk with your specific setup and Solasta? Maybe someone else might have more insight to share on the subject.