The parabolic component conjecture for nilpotent commuting varieties

Let $G$ be the relevant general linear group, let $P$ be the parabolic subgroup corresponding to the partition $[m,m]$ when $n=2m$ or $[m+1,m]$ when $n=2m+1$, and let $\mathfrak{u}_P$ be the Lie algebra of its unipotent radical. Set

$$V_P=G\cdot\mathfrak{u}_P^r.$$

Then $V_P$ is a closed irreducible subvariety of $C_r(\mathcal{N}_n)$.

Parabolic component conjecture. The variety $V_P$ is an irreducible component of $C_r(\mathcal{N}_n)$ for all $n,r\ge 4$.

The claim would identify a natural component arising from the Richardson parabolic construction. The supplied text gives dimension estimates and says that the assertion is claimed for most values, but does not establish it or state its resolution.