I used coconut oil for 20 minutes each morning. When I saw the results and read of the benefits of using different oil, a began using canola oil for a second pull. So I now do two 20 minute pulls, back-to-back, first canola and then coconut.
For a while I cut back to a 10 minute pull, due to lack of time, and the KP started to come back. I resumed the 20 minute duration, and it diminished. That was proof for me that it was the OP that made the difference.
Interestingly, some on the KP forum use coconut oil topically to remedy dry skin. That I chose coconut oil was a coincidence, but I find it interesting that the oil that worked for me for OP was the same one some of them had limited success with in diminishing KP when rubbed on the affected areas.
Good luck. Let me know if this makes a difference.