Hausel's equivariant K-theory 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 condition that its points are pairs $(h,x)\in \mathrm G\times X$ with $hx=x$, and let $\mathrm G$ act on it by

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

The invariant-function ring $\mathbb C[\operatorname{Fix}_{\mathrm G}(X)]^{\mathrm G}$ is an algebra over $\mathbb C[\mathrm G]^{\mathrm G}\cong K^0_{\mathrm G}(\operatorname{pt})\otimes\mathbb C$, and it should be isomorphic to equivariant algebraic K-theory $K^0_{\mathrm G}(X)\otimes\mathbb C$, compatibly with these structure maps:

\begin{tikzcd} \mathbb C[\operatorname{Fix}_{\mathrm G}(X)]^{\mathrm G} \arrow{r}{\cong} & K_{\mathrm G}^0(X)\otimes \mathbb C \\ \mathbb C[\mathrm G]^{\mathrm G} \arrow{r}{\cong} \arrow{u}& K_{\mathrm G}^0(\operatorname{pt})\otimes \mathbb C \arrow{u}. \end{tikzcd}

This conjecture proposes a fixed-point-scheme model for equivariant algebraic K-theory, extending the preceding analogy between equivariant cohomology and invariant functions on the Lie algebra. Its validity is not resolved in the supplied text; the paper states it as a conjecture based on an earlier theorem.