Haiman's h-positivity conjecture for Kazhdan–Lusztig characters

Let $S_n$ be the symmetric group, let $H_n$ be its Hecke algebra, and let $C'_w$ denote the Kazhdan–Lusztig basis element associated to $w\in S_n$. For a partition $\lambda\vdash n$, write $s_\lambda(x)$ for the Schur symmetric function, and define the dual Frobenius character by

$$\operatorname{ch}(a)=\sum_{\lambda\vdash n}\chi^\lambda(a)s_\lambda(x).$$

A symmetric function is $h$-positive when its coefficients in the complete homogeneous basis $\\{h_\lambda\\}$ are polynomials in $q$ with non-negative coefficients.

Haiman's conjecture. For any $w\in S_n$, the dual Frobenius character $\operatorname{ch}(q^{\frac{\ell(w)}{2}}C'_w)$ is $h$-positive.

This conjecture connects Kazhdan–Lusztig character positivity with the positivity of chromatic symmetric and quasisymmetric functions of indifference graphs. The supplied text does not state a resolution.