Lots of good advice. Your TIR on the test is very linear with distance, so the rod looks good at this position. I assume the max min in the runout occurred at the same phase of chuck rotation? So, another test suggestion. After cleaning, but before reworking the chuck I would do a simple test to check the chuck jaws. 1) Measure the run out as a function of rotation angle phase, not just TIR. Unclamp the bar and rotate it ~ 120 degrees or so in a 3 jaw chuck. Keeping track of the phase angle of the chuck as well as the phase of the bar. Remeasure. Are the max points and min points at the same phase angle or did they change with the jaw clamping? Repeat at other angles to be sure. In addition to uneven jaws, I would think that small amounts of debris at the jaw can make it clamp at an angle other than on axis. Likewise, a bar which has a bump, debris, or bend at the jaws.
2) If you have not done so recheck your lathe bed for level. Really the first think to do. I do not expect my PM1440GT to remain level over 2 years. My concrete floor sets on clay and as it gets wet and dry it moves!
Good luck.