If the strings are binding in the nut slots, the tuning will not stay accurate. You can test this by plucking an open string, and then pressing on that string between the nut and the tuner. It will go sharp when you prss on it, but should return to accurate pitch when you release it. If it stays a bit sharp, get a good tech to open up the string slots a little bit. A touch of pencil lead (graphite) in the slot may also help to lubricate the strings.