Characteristic-cycle isomorphism conjecture for top category O

Let $\mathfrak{A}(\mathbf v,\mathbf w)$ be the attracting locus above, let $d$ denote its expected relevant dimension, let $\varphi$ be an integral parameter, and let $\mathcal O_{\operatorname{top}}(\mathbf v,\mathbf w)$ be the quotient of $\mathcal O(\mathsf{CB}_{\varphi}(\mathbf v,\mathbf w))$ by objects of Gelfand–Kirillov dimension less than $d$. The characteristic-cycle map sends a class in this quotient to its top Borel–Moore homology class.

Characteristic-cycle conjecture. One has

$$\dim \mathfrak{A}(\mathbf v,\mathbf w)=d,$$

and, for any integral $\varphi$, the map

$$\bigoplus_{\mathbf v}K_{\mathbb C}(\mathcal O_{\operatorname{top}}(\mathbf v,\mathbf w))\longrightarrow \bigoplus_{\mathbf v}H^{BM}_{2d}(\mathfrak{A}(\mathbf v,\mathbf w))$$

induced by characteristic cycles is an isomorphism of vector spaces.

The conjecture would identify the representation obtained from the top quotient of category $\mathcal O$ with the top Borel–Moore homology representation. The source presents it as the remaining step in reducing the preceding generalized geometric Satake conjecture and gives no resolution.