Primality conjecture for odd-size jet pfaffian ideals

Let $r$ and $k$ be positive integers, and let $I^{2r+1,k}_r$ denote the jet pfaffian ideal associated with the ideal of size-$2r$ pfaffians of a generic $(2r+1)\times(2r+1)$ skew-symmetric matrix and its $k$-jet scheme. Primality conjecture. For all positive integers $r$ and $k$, the jet pfaffian ideals $I^{2r+1,k}_r$ are prime.

The corresponding classical pfaffian ideals are prime, and the preceding theorem shows that the associated jet pfaffian varieties are irreducible of codimension $3k$. The conjecture asks for the stronger property of primality for the general jet pfaffian ideals; the paper gives computational evidence in a small case.