The unique-sign conjecture for Kazhdan–Lusztig two-sided cells

Let $W$ be a finite Coxeter group, let ${\mathfrak{c}}$ be a two-sided cell of $W$, let $\Lambda$ index the irreducible representations $E^\lambda$ of $W$, and write $E^{\lambda} \sim_L {\mathfrak{c}}$ for the relevant left-cell relation. For each $w\in W$ and $\lambda\in\Lambda$, let $c_{w,\lambda}$ be the associated coefficient.

Unique-sign conjecture. There is a unique $\lambda_0\in\Lambda$ such that $E^{\lambda_0}\sim_L {\mathfrak{c}}$ and $c_{w,\lambda_0}$ has the same sign for all $w\in {\mathfrak{c}}$ such that $c_{w,\lambda}\neq 0$.

This conjecture asserts a canonical sign choice associated with every Kazhdan–Lusztig two-sided cell. The supplied excerpt gives no surrounding definitions or evidence of a resolution, so its precise representation-theoretic setting and current status should be checked against the source.