Hey, i own a bunch of les pauls, i have encountered this problem a bunch of times and finally figured out how to fix it.
Get some 1000 and 2000 grit sandpaper. the next time you change your strings gently run the 1k then 2k paper through the slot until it feels smoth, make sure you get the top and bottom edge... from a string perspective its just where the string lays on the nut itself.. I bought a jewelers loop and use it to check out the slots on guitars i buy, the slots are sometimes sloppy. Get a magnifying glass or something and check it out, dont spend money on string managers or any of that nonsense. That less paul nut is fine it just needs a bit of love and attention. 99.9999% of tuning problems on les ppauls is just the nut. I usually get a bone nut installed right away except on the 58's i have, i leave those as is.