Artin–Hochster conjecture on the Cohen–Macaulayness of the commuting scheme

Given $n\in\mathbb Z_{\ge 0}$, let the commuting scheme $\mathfrak C_n$ be the affine scheme of pairs of commuting $n\times n$ complex matrices:

$$\mathfrak C_n=\{(A,B)\in\mathfrak{gl}_n(\mathbb C)^2:[A,B]=0\}.$$

Artin–Hochster conjecture. The scheme $\mathfrak C_n$ is Cohen–Macaulay.

The commuting scheme is known to be irreducible, of dimension $n(n+1)$, and smooth in codimension one. Cohen–Macaulayness remains the conjectural property attributed generally to Artin and Hochster in 1982.