SNP conjecture for matrix Schubert coefficient supports

Fix a permutation $w$. Identify each partition-tuple $(\underline\lambda|\underline\mu)$ with a point in ${\mathbb R}^{m+n}$, and let

$${\mathcal F}_w:=\{(\underline\lambda|\underline\mu)\mid c^w_{\underline\lambda|\underline\mu}\neq 0\}.$$

A lattice point means a point of the ambient lattice ${\mathbb Z}^{m+n}$. SNP conjecture. Every lattice point $(\underline\lambda|\underline\mu)$ in the convex hull of ${\mathcal F}_w$ belongs to ${\mathcal F}_w$:

$$(\underline\lambda|\underline\mu)\in\operatorname{conv}({\mathcal F}_w)\cap{\mathbb Z}^{m+n}\quad\Longrightarrow\quad(\underline\lambda|\underline\mu)\in{\mathcal F}_w.$$

This is a normality or saturation-type property for the coefficient support. The source presents it as a third conjecture and does not state a general proof or disproof.