K-theoretic McKay conjecture for tautological exterior powers

Let $X$ be a scheme of dimension at least $2$, let $K,L$ be line bundles on $X$, and let $p_i:X^n\to X$ be the projections. Set

$$U=\bigoplus_{i=1}^n p_i^*K,\qquad V=\bigoplus_{i=1}^n p_i^*L.$$

Let $\rho:X^{[n]}\to X^{(n)}$ be the Hilbert–Chow morphism, and let $\pi:[X^n/\mathfrak{S}_n]\to X^{(n)}$ be the projection from the quotient stack. Tautological McKay conjecture. The K-theoretical pushforward satisfies

$$\rho_*\left(\Lambda_{-u}\big((K^{[n]})^*\big)\otimes\Lambda_{-v}\big(L^{[n]}\big)\right) \cong \pi_*\left[(\Lambda_{-u}U^*\otimes\Lambda_{-v}V)/\mathfrak{S}_n\right].$$

This extends the structure-sheaf pushforward to exterior powers of tautological bundles. It is presented as a conjectural higher-dimensional McKay-type identity; the paper does not establish it in general.