Geiss-Leclerc-Schröer conjecture on characteristic cycles and semi-canonical bases

Let $(I,Q)$ be a type $A$ quiver with orientation $i\rightarrow i+1$, let $V$ be an $I$-graded vector space, and let $S$ and $S'$ be $G_V$-orbits in the representation variety $E_{V,\Omega}$. Write $T^*_{S}E_{V,\Omega}$ for the conormal bundle to $S$, let $\phi_S$ be the semi-canonical basis element associated with $\overline{T^*_{S}E_{V,\Omega}}$, and define $m_{S',S}$ by

$$g_S=\sum_{S'}m_{S',S}\phi_{S'}.$$

If

$$CC(\operatorname{IC}(\overline{S},\mathbb{C}))=[T^*_{S}E_{V,\Omega}]+\sum_{S'\subseteq\overline{S}}n_{S',S}[T^*_{S'}E_{V,\Omega}],$$

then $Eu$ denotes the Kashiwara-Schapira morphism from Lagrangian cycles to constructible functions. Geiss-Leclerc-Schröer conjecture. For every orbit $S$,

$$Eu([T^{*}_{S}E_{V, \Omega}])= (-1)^{{\rm \dim} S} \phi_{S},$$

equivalently,

$$m_{S', S} = (-1)^{{\rm \dim} S' - {\rm \dim} S} n_{S', S}.$$

This conjecture identifies the coefficients relating the canonical and semi-canonical bases with the multiplicities of characteristic cycles. The paper studies it for type $A$ quivers and develops a strategy for the general case; the supplied text gives no resolution of the conjecture in full generality.