The irreducibility implication for ordinary and nilpotent commuting varieties

Let $k$ be an algebraically closed field, let $\mathfrak{gl}_n$ be the Lie algebra of $n\times n$ matrices over $k$, and let $\mathcal{N}_n$ be the variety of nilpotent matrices. For $r\ge 2$, let $C_r(V)$ denote the variety of pairwise commuting $r$-tuples in $V$.

Irreducibility implication. If $C_r(\mathfrak{gl}_n)$ is irreducible, then $C_r(\mathcal{N}_n)$ is irreducible.

This is presented as an analogy between ordinary and nilpotent commuting varieties. The source does not provide a proof or resolution of the assertion.