Advice for tuning please

Guys i would like to know if any of you have an idea, what i could do on my Gibson Les Paul that have a bad tuned G string.

The G string as i have heard is often a problem on Gibson guitars?..

Do you have any tips on things i could try to get it to tune better. The intonation screw is almost all the way back which i also find pretty strange (but it tells me that the intonation is perfect when i test that).

What i have tried:

1. I have tried to put graphite down the nut slot which helped actually helped with the string not ''jumping'' when i tune or play.

2. The nut slot on the G string is cut so that the G string lays a bit lower than the rest of the strings since i was told that it could help. An open E chord now sounds beautiful (mostly on guitars an open E chord don't sound too good to be honest) but when i reach around the 3-9 frets is not really spot on with the notes?

3. I have messed a bit with the truss rod (only tiny adjustments ofcourse), and the guitarneck is now as straight as i can get it (maybe i tiny bit too much curve backwards). Can that affect the tuning of the strings?

Do you guys have any other things i should try?

I have the same problem on my Epiphone Les Paul 100. Every time I pick it up, it is out of tune on the G String. The other five strings stay perfectly in tune, even through temperature changes and stuff that should knock it out of tune, but the G String always goes out of tune. If anyone knows a solution, please let us know.

As a troubleshooting step, tune the G string around 10 cents flat and see how it sounds. How 'bout 10 cents sharp?

How is the action set? The strings will be higher above the fretboard the higher up the neck you play. If the action is high, you could be pulling the strings out of tune by simply playing.

What gauge strings are you using? 9s and lighter can easily be made to sound sour by fretting with excessive pressure. Try playing with a lighter touch, or try 10s.

Make sure the pickups aren't too high. I've had weird intonation problems from having a bridge pickup to close to the strings on a Strat.

Check for loose hardware. A buzz somewhere in the G string hardware could be making it sound wonky.

Some people are really sensitive to the compromises in equal temperament tuning. Maybe you're one of them.

I had a professional set it up for me, so the action is pretty low. especially on the G string where he cut the nut a little further down compared to the other strings, so that it tunes better.

To me i think it sounds a tad better if the string is tuned a TINY TINY TINY bit flat, but it's still not totally in order to my ears.
I think ill take it to one of our big music stores in the city where i could maybe get some people to play it and hear what they think about the tuning, if they can even hear it.