The Littlewood–Richardson conjecture for Hermitian symmetric spaces

Let $G_{\Bbb R}/K_{\Bbb R}$ be an irreducible Hermitian symmetric space of rank $r$, let $X=\frak p^+$, and let $\Lambda$ be the set of partitions of length at most $r$. For $\lambda\in\Lambda$, let $S_\lambda$ be the corresponding nonzero $K$-isotypic component of ${\Bbb C}[X]$, and let $c^\nu_{\lambda\mu}$ denote the Littlewood–Richardson coefficient for $GL_r({\Bbb C})$. The Littlewood–Richardson conjecture. For the $K$-action on $X$,

$$S_\nu\subseteq S_\lambda\cdot S_\mu \quad\text{if and only if}\quad c^\nu_{\lambda\mu}\ne 0,$$

where $\lambda,\mu,\nu\in\Lambda$. Ruitenburg made this conjecture for the tube-type cases and observed it for $X={\Bbb C}^p\otimes({\Bbb C}^p)^*$; the source gives no general resolution.