Kivinen's fixed-point and equivariant K-theory conjecture for symbolic varieties

Let $\mathfrak{g}$ be a reductive Lie algebra with Weyl group $W$, let $\mathfrak{C}_W$ be the set of two-sided cells in $W$, and let $\mathbf{a}:\mathfrak{C}_W\to\mathbb{N}$ be Lusztig's function. Define $\mathbf{A}(\lambda)=\mathbf{a}(w_0\lambda)$, and let $X_{\mathfrak{g},\mathrm{symb}}$ carry its natural $(\mathbb{C}^*)^2$-action. If $\delta_\lambda$ is the skyscraper sheaf at the fixed point corresponding to $\lambda$, then in $K^0_{(\mathbb{C}^*)^2}(X_{\mathfrak{g},\mathrm{symb}})$ one can consider its twist by $\mathcal{O}(1)$. Kivinen's fixed-point and equivariant K-theory conjecture. There is a bijection

$$X_{\mathfrak{g},\mathrm{symb}}^{(\mathbb{C}^*)^2}\leftrightarrow\mathfrak{C}_W,$$

and

$$[\delta_\lambda\otimes\mathcal{O}(1)]=q^{\mathbf{a}(\lambda)}t^{\mathbf{A}(\lambda)}[\delta_\lambda].$$

The first part follows in type $A$ and is proved in types $B$ and $C$ in the appendix; the full conjecture remains open in general.