Lam–Postnikov–Pylyavskyy's alcoved-polytope conjecture for sl_n character products

Let $\alpha_{ij}=e_i-e_j$ be the roots of the type $A$ root system. An alcoved polytope is a polytope whose faces lie in hyperplanes $\langle\alpha_{ij},\tau\rangle=m$ for $m\in\mathbb Z$. For weights $\lambda$ and $\mu$, let $P_{\lambda,\mu}$ be the minimal alcoved polytope containing them. Let $\nu$ and $\rho$ be another pair of weights. Lam–Postnikov–Pylyavskyy's conjecture. If

$$\lambda+\mu=\nu+\rho$$

and $\nu,\rho\in P_{\lambda,\mu}$, then $\chi_{\nu}\chi_{\rho}-\chi_{\lambda}\chi_{\mu}$ is $\chi$-nonnegative. This would give a sufficient condition for $V_{\lambda}\otimes V_{\mu}$ to be a submodule of $V_{\nu}\otimes V_{\rho}$, equivalently for the stated character difference to be $\chi$-nonnegative, in the equal-sum case. The source provides no evidence of a resolution, so the conjecture is recorded as open.