Orbit-product decomposition conjecture with a nontrivial stabilizer

Let $O(\lambda)$ and $O(\mu)$ be orbits with $\lambda\ne 0$ and $\mu\ne 0$. Suppose a stabilizer subgroup of $\mu$ is generated by the reflections $r_{i_1},\ldots,r_{i_s}$ corresponding to simple roots $\alpha_{i_1},\ldots,\alpha_{i_s}$. If, for each $w\in W/W_\lambda$, one has

$$\langle w\lambda+\mu,\alpha_j\rangle>0\quad\text{for all }j\notin\{i_1,i_2,\ldots,i_s\},$$

then the product decomposes as a union of the corresponding orbits.

Orbit-product decomposition conjecture. Under these hypotheses,

$$O(\lambda)\otimes O(\mu)=\bigcup_{w\in W/W_\lambda}O(w\lambda+\mu).$$

This is one of the paper's conjectures about decomposing products of orbits. The supplied text gives no evidence that it has been resolved.