Involutivity and radicality of Pfaffian ideals for Severi strata

Let $S$ be the symplectic manifold with symplectic form $\Omega$, let $\chi$ denote the relevant matrix of logarithmic vector fields, and let $\operatorname{Pf}_{2k}(\chi^t\Omega\chi)$ be the ideal generated by the $2k$-Pfaffians of the matrix $\chi^t\Omega\chi$. For $k=1,2,\ldots,\delta$, the Pfaffian-ideal conjecture.

$$\operatorname{Pf}_{2k}(\chi^t\Omega\chi)\text{ is involutive and radical.}$$

Involutivity is suggested by the coisotropic nature of the Severi strata, while radicality is needed to pass from the geometric coisotropic statement to the corresponding ideal-theoretic assertion. The source gives no resolution of either claim.