Characteristic-cycle image conjecture for quantized quiver varieties

Let $Q$ be a quiver without loops, let $\lambda$ be a parameter, and let $\mathcal{A}_\lambda(v)$ be the corresponding quantized quiver-variety algebra. Let $\mathsf{CC}^\lambda$ be the characteristic-cycle map from the Grothendieck group of finite-dimensional modules, and let $L_\omega$ be the relevant integrable representation of $\mathfrak{g}(Q)$. Define $\mathfrak{a}\subset\mathfrak{g}(Q)$ to be generated by the Cartan subalgebra and the real-root spaces $\mathfrak{g}(Q)_\beta$ for which $\sum_{i\in Q_0}b_i\lambda_i\in\mathbb{Z}$ when $\beta=\sum_{i\in Q_0}b_i\alpha^i$, and let $L_\omega^\mathfrak{a}$ be the $\mathfrak{a}$-submodule generated by the spaces $L_\omega[\sigma\omega]$ for $\sigma\in W(Q)$. Characteristic-cycle image conjecture. One has

$$\operatorname{Im}\mathsf{CC}^\lambda=L_\omega^\mathfrak{a}.$$

This conjecture describes the characteristic cycles of finite-dimensional modules and, in particular, predicts their number. The source states that the integral-parameter case follows from Webster's construction, while the general case was planned for future work.