...
I think that conclusively disproves your conjecture, by counterexample.
Week 3 - Dynamic Programming in Impartial Games
Open Problem: Cram in odd x odd case:
In the even x even board, Cram is 2nd player win, if you take two reflections across the axes. In the even x odd case, it's a 1st player win if the player plays in the middle of the board. What happens in the odd x odd board?
Erik's Conjecture: in any 3 x odd board, besides the 3 x 3 case, moving in the middle as the first move will give the 1st player a win.