restring the guitar. only leave 2 inches of string above the tuner. the string should wrap around the tuner with only 2 or 3 turns. more than that will cause tuning issues.
cut the 4th string excess lenght. take that piece of excess string and run it up and down the 3rd string slot pressing sidesays, not down. you don’t want to make the nut slot deeper, just wider. Make sure your get both sides. go slow so you don’t widen the slot too much. Also run the string a few times through the bridge saddle.
the guitar will sound more open and also stay in tune. I have done this in all my les pauls and esp eclipses, no tuning problems.