Unimodality and log-concavity conjecture for Mather polynomials

Let $X=G/P$ be a cominuscule space and let $w\in W^P$. Write

$${c_{\operatorname{Ma}}}(X_w^P)=\sum_v a_{w,v}[X_v^P]$$

and define the Mather polynomial by

$$M_w(x)=\sum_v a_{w,v}x^{\ell(v)}.$$

Unimodality and log-concavity conjecture for Mather polynomials. The polynomial $M_w$ has strictly positive coefficients and is unimodal. If in addition $G/P=\operatorname{Gr}(k,n)$, then $M_w$ is log concave. These properties are motivated by substantial calculations in all Lie types; the source gives no resolution status.