Path being a Hamiltonian path through the centers of the orthogonal sudoku grid cells of the odd numbers. Standard sudoku rules imply 5 unique odd numbers in each row, column, and box.
The uniqueness constraint seemingly has to do with the clustering of cells of degree 3 or more in the middle and 2 or more around the endings.
Update: It is possible with one restricted endpoint (r1c5/r1c7/r2c6), it has 3 permutations without it. I found it manually, try it yourself here: f-puzzles
Outdated: The best I have found is one with 4 permutations, see below.
If it isn't possible then is it possible with restricted endpoints? Restricting the above example to a 1 or 3 ending in the top left box reduces its permutation count to 2.
