Hausel's K-theoretic conjecture for fixed-point schemes

Let $\mathrm G$ be a principally paired group acting regularly on a smooth projective variety $X$. Define the fixed-point scheme $\operatorname{Fix}_{\mathrm G}(X)$ by the pullback of the diagonal map $X\to X\times X$ along the map $\mathrm G\times X\to X\times X$, $(g,x)\mapsto (gx,x)$. Let $\mathrm G$ act on $\operatorname{Fix}_{\mathrm G}(X)$ by

$$g\cdot (h,x)=(ghg^{-1},gx).$$

Hausel's K-theoretic conjecture. The ring of invariant functions is, as an algebra over $\mathbb C[\mathrm G]^{\mathrm G}\cong K^0_{\mathrm G}(\operatorname{pt})\otimes\mathbb C$, isomorphic to equivariant algebraic K-theory:

$$\mathbb C[\operatorname{Fix}_{\mathrm G}(X)]^{\mathrm G}\cong K^0_{\mathrm G}(X)\otimes\mathbb C.$$

Equivalently, this isomorphism should fit into the natural commutative diagram with the corresponding isomorphism $\mathbb C[\mathrm G]^{\mathrm G}\cong K^0_{\mathrm G}(\operatorname{pt})\otimes\mathbb C$. The conjecture proposes a fixed-point-scheme realization of equivariant algebraic K-theory, extending the preceding realization of equivariant cohomology by zero schemes; its resolution is not given in the source.