Semigroup conjecture for matrix Schubert coefficient supports

Fix a permutation $w$, and let

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

Here $\underline\lambda$ and $\underline\mu$ are partition-tuples and $c^w_{\underline\lambda|\underline\mu}$ is the associated coefficient. Semigroup conjecture. The set ${\mathcal F}_w$ is a semigroup: if

$$c^w_{\underline\lambda|\underline\mu}\neq 0\quad\text{and}\quad c^w_{\underline\lambda'|\underline\mu'}\neq 0,$$

then

$$c^w_{\underline\lambda+\underline\lambda'|\underline\mu+\underline\mu'}\neq 0.$$

For Littlewood–Richardson coefficients this follows from the polytopal rule, and the property has been computationally verified in the finite cases listed in the source. Its validity in general remains open.