The nilpotent centralizer conjecture for generalized punctual Hilbert schemes

Let $\mathfrak{g}$ be a semisimple Lie algebra of rank at least $3$, let $A\in\mathfrak{g}$ be nilpotent, and write $Z(A)$ for its centralizer and $\operatorname{rk}\mathfrak{g}$ for the rank. Nilpotent centralizer conjecture. There is $B\in Z(A)$ nilpotent such that

$$\dim Z(A,B)=\operatorname{rk}\mathfrak{g},$$

i.e. $[(A,B)]\in\operatorname{Hilb}_0(\mathfrak{g})$. This should be true for a generic element $B\in Z(A)$. The claim would ensure that every nilpotent conjugacy class occurs in the zero-fiber, but the paper leaves it conjectural.