It is rare to find a Les Paul without a tuning stability issue. Certainly my Gibson Traditional Pro has the same issue. Replacing tuners is always a good idea, but this problem is usually a friction on the nut thing, due to string angle over the nut. Graphite can work, but if it’s a bone or white nut, it can look terrible. In those cases, get some nut sauce.
https://www.amazon.com/Big-Bends-Luber-Ultra-Applicator/dp/B000SHQ1QC/ref=sr_1_1?keywords=nut+sauce&qid=1550065121&s=gateway&sr=8-1