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With the strings off (or if you're very good at getting oil precisely where you need it), apply a small drop of light oil where the tuner post contacts the metal ferrule when the string is under tension. See small red dots in the pic--that's where the drop of oil goes. If you don't have a needle oiler, apply a drop to the point of a wooden toothpick and use that to apply the oil. A little oil is better than a lot.
DO NOT get any oil in the portion of the post where the strings are wrapped--because that can make them slip off the post.