The multiplicity-free product conjecture for Hermitian symmetric spaces

Let $G_{{\Bbb R}}/K_{{\Bbb R}}$ be an irreducible Hermitian symmetric space of non-compact type, let $\frak g=\frak k\oplus\frak p^+\oplus\frak p^-$, let $K$ be the complexification of $K_{\Bbb R}$, and let $X=\frak p^+$. Write $S_\lambda$ for the irreducible $K$-isotypic component of ${\Bbb C}[X]$ indexed by $\lambda\in\Lambda$, and $V_\lambda$ for the corresponding irreducible $K$-module. The multiplicity-free product conjecture. For the $K$-action on $X$,

$$S_\nu\subseteq S_\lambda\cdot S_\mu \quad\text{if and only if}\quad V_\nu\hookrightarrow V_\lambda\otimes V_\mu,$$

where $\lambda,\mu,\nu\in\Lambda$. This proposes that nonzero products of isotypic components are detected exactly by tensor-product containment; the source presents it as one of two conjectures, with no resolution supplied.