Support conjecture for Littlewood–Richardson coefficients of interpolation polynomials

Fix partitions $\mu$ and $\nu$. For each family $\mathcal F$, let

$$S_{\mu\nu}^{\mathcal F}=\left\{\lambda:c_{\mu\nu}^{\lambda,\mathcal F}\neq0\right\}$$

and let $M_{\mu\nu}$ be the set of partitions $\lambda\supseteq\mu,\nu$ for which there exists a Molev tableau of type $(\lambda,\mu,\nu)$. Support conjecture. For $\mathcal F={A\mathrm{J}}, {B\mathrm{J}}, {A\mathrm{M}}$ and ${B\mathrm{M}}$, all the sets $S_{\mu\nu}^{\mathcal F}$ are equal to $M_{\mu\nu}$. The equality is motivated by the known support description for double Schur polynomials via Molev tableaux; its status for all four interpolation-polynomial families is not resolved in the supplied text.