K-theoretic McKay pushforward conjecture for the Hilbert–Chow morphism

Let $X$ be a smooth projective scheme, let $X^{(n)}=X^n/\mathfrak{S}_n$ be its $n$th symmetric product, and let

$$\rho:X^{[n]}\longrightarrow X^{(n)}$$

be the Hilbert–Chow morphism. K-theoretic McKay conjecture. The K-theoretical pushforward satisfies

$$\rho_*\mathcal{O}_{X^{[n]}}=\mathcal{O}_{X^{(n)}}$$

in $K_0(X^{(n)})$.

For smooth projective surfaces this follows from the cited results, while the higher-dimensional statement remains open. It predicts the expected structure-sheaf pushforward under the Hilbert–Chow resolution.