I don't have time to go further but you can at least add all these red and purple crosses without guessing :
View attachment 322287
Start with the bottom left red cross and check what would happen if there were a black spot there, you'll see that you quickly get to a contradiction. Then work your way up with the same logic, and then do the same on the second column. The two purple crosses are quite harder and will require you to visualise several steps before getting to the contradiction, but if I didn't make a mistake they should be guaranteed too.
I assume by continuing this "ad absurdum" kind of reasoning you will end up unlocking the puzzle without requiring a guess.