Haiman's positivity and unimodality conjecture for symmetric groups

Let $S_n$ be the symmetric group, let $\boldsymbol{\mu \vdash n}$, and let $\operatorname{mon}_{\mu}$ be the class function whose Frobenius characteristic is the monomial symmetric function indexed by $\mu$. For $z\in S_n$, define

$$\alpha_{\mu}^{z}(\mathsf{v})=\operatorname{mon}_{\mu,\mathsf{v}}(c_z)\in\mathbb{Z}[\mathsf{v}^{\pm1}].$$

Haiman's conjecture. For every $z\in S_n$ and $\mu\vdash n$, the nonzero coefficients of $\alpha_{\mu}^{z}$ are all positive and form a unimodal sequence.

This conjecture strengthens the known positivity for irreducible characters of the symmetric group and is the motivating precedent for extending positivity and unimodality to other Weyl groups.