Universal spectral data morphism conjecture

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 ${\mathfrak g}$. There is a natural action of $G$ on ${\mathfrak C}^d_G$ and a quotient ${\mathfrak t}^d\sslash W$. Universal spectral data morphism conjecture. There exists a $G$-invariant morphism

$$\operatorname{sd}:{\mathfrak C}^d_G\to {\mathfrak t}^d\sslash W$$

that makes the characteristic quotient diagram commute. This is presented as a weaker form of the conjecture that ${\mathfrak C}^d_G\sslash G$ is reduced and normal.