Chevalley restriction conjecture for the commuting quotient

Let $G$ be a reductive group with Lie algebra ${\mathfrak g}$, let ${\mathfrak t}$ be a Cartan subalgebra, let $W$ be the Weyl group, and let ${\mathfrak C}^d_G$ be the scheme of commuting $d$-tuples in ${\mathfrak g}$. The inclusion ${\mathfrak t}^d\to{\mathfrak C}^d_G$ induces a morphism

$${\mathfrak t}^d\sslash W\to {\mathfrak C}^d_G\sslash G.$$

Chevalley restriction conjecture. This morphism is an isomorphism. Equivalently, the categorical quotient ${\mathfrak C}^d_G\sslash G$ is reduced and normal. The source records this as a conjecture and supplies no resolution.