"Data race which doesn't seem to modify the data" is actually often not a false positive, they're called "Benign data races" and well, go read the literature, it's often difficult to be sure they're truly "benign" and if there's a subtle reason why they aren't that's now a crazy hard to spot bug in your program.

So yeah, just fix 'em. Most of them you dodged a bullet, and even when you didn't it will now be easier for the next person to reason about the software.