Multiplicity-one conjecture for products of Weyl-group orbits

Let $O(\lambda)$ and $O(\mu)$ be orbits with $\lambda\ne 0$ and $\mu\ne 0$, and let $\mu$ be strictly dominant. If, for some $w\in W$, the element $w\lambda+\mu$ is strictly dominant, then the multiplicity of $O(w\lambda+\mu)$ in $O(\lambda)\otimes O(\mu)$ is $1$.

Multiplicity-one conjecture. Under these hypotheses, the multiplicity of $O(w\lambda+\mu)$ in $O(\lambda)\otimes O(\mu)$ is $1$.

This conjecture concerns multiplicities in decompositions of products of Weyl-group orbits. The supplied text does not state whether it has been proved or disproved.