Good move - persimmons love to starve out grafts, you need to watch them carefully. Those particular shoots are low enough relative to the grafts that it might end up being OK, but I have had cases like there where the grafts seemed to start but then get out-competed by the other growth and never get going. This is on persimmons, other fruit trees think differently.
Speaking of persimmons, it looks like my 12-year Hachiya did not make it through the last winter. Thats sad, it was a big beautiful and productive tree.