Lam–Postnikov–Pylyavskyy alcoved-polytope Schur-order conjecture

Let $\rho,\mu,\nu,\delta$ be partitions, and let $\chi_\lambda$ denote the corresponding $\operatorname{SL}_n$ character. Suppose

$$\mu+\nu=\rho+\delta$$

and that $\mu$ and $\nu$ belong to the minimal alcoved polytope containing $\rho$ and $\delta$. Write $A\geqslant_s B$ when $A-B$ is Schur positive. Lam–Postnikov–Pylyavskyy conjecture. Under these conditions,

$$\chi_\mu\chi_\nu\geqslant_s\chi_\rho\chi_\delta.$$

The source attributes this to an unpublished conjecture of Lam, Postnikov, and Pylyavskyy; it also states that the corresponding symmetric-function conjecture implies a linear-progression strong Schur-log-concavity claim. No resolution is given.