I searched all the fully symmetric dispositions of 17 givens on a 9x9 grid, with a program I wrote, and found no valid problem : every disposition of values had more than one solution. The search used the spare cycles of half a dozen of PCs for around three month.

It is not really a proof, as the validity of my search program is unproven. I, at least, am convinced this (negative) result is exact.

My 2 c.