Linear independence of cell modules for distinguished nilpotents

Let $e$ be a distinguished nilpotent element, so that $A_e=Q_e$, and let $c$ be the corresponding two-sided cell. Let $J_c$ be identified with $K_0(\operatorname{Sh}^{Q_e}(Y_e\times Y_e))$, and let $\mathcal{O}$ range over the $Q_e$-centrally extended orbits appearing in $Y_e$. Cell-module independence conjecture. In the Grothendieck group of the category of $J_c$-modules, the classes of the cell modules

$$K_0(\operatorname{Sh}^{Q_e}(Y_e\times \mathcal{O}))$$

for non-isomorphic $\mathcal{O}$ are linearly independent. This conjecture concerns the independence of the different left cell-module classes and is motivated by the analogous result for finite Weyl groups; the supplied passage gives no resolution.