Generalized complete-intersection conjecture for Jordan-type loci

Let $Q$ be a stable partition of length $\ell$, let $J_Q$ be a Jordan matrix of type $Q$, and let $\mathcal N_{J_Q}$ be the variety of nilpotent matrices commuting with $J_Q$. For a partition $P$ with $\mathfrak D(P)=Q$, let $W^Q_P$ denote the corresponding Jordan-type locus. Generalized Jordan-type-locus conjecture. The closure of $W^Q_P$ in $\mathcal N_{J_Q}$ is an irreducible complete intersection of codimension $\ell(P)-\ell$, with defining equations of degree at most $\ell$. The source presents this as a proposed generalization of an earlier theorem, and gives no resolution here.