Chevalley restriction conjecture for commuting schemes

Let $k$ be a field of characteristic $0$, let $G$ be a reductive algebraic group over $k$ with Lie algebra $\mathfrak g$, and fix a maximal torus $T\subset G$ with Lie algebra $\mathfrak t$ and Weyl group $W$. For $n\in\mathbb N$, let

$$\mathfrak C_{\mathfrak g}^n=\Bigl\{(x_1,\dots,x_n)\in\mathfrak g^n\ \big|\ [x_i,x_j]=0\text{ for all }1\le i,j\le n\Bigr\}$$

be the $n$-fold commuting scheme, and write $\mathfrak C_{\mathfrak g}^n\sslash G=\operatorname{Spec}\left(k[\mathfrak C_{\mathfrak g}^n]^G\right)$. The inclusion $\mathfrak t\hookrightarrow\mathfrak g$ induces

$$\vartheta:k[\mathfrak C_{\mathfrak g}^n]^G\longrightarrow k[\mathfrak t^n]^W.$$

Chevalley restriction conjecture for commuting schemes. The morphism $\vartheta$ is an isomorphism. This asserts that invariant functions on commuting $n$-tuples are completely determined by restriction to tuples in a fixed maximal torus. The claim concerns the invariant-theoretic structure of commuting schemes and remains unresolved here.