PRV conjecture for the fusion product

Let $\mathfrak{g}$ be a simple Lie algebra, let $P_{\ell}$ be the set of level-$\ell$ dominant weights, let $W_{\ell}$ be the affine Weyl group, and let $V(\lambda)$ denote the irreducible $\mathfrak{g}$-module of highest weight $\lambda$. For $\lambda,\mu\in P_{\ell}$ and $w\in W_{\ell}$, write $\overline{\lambda+w\mu}^{F}$ for the element of $P_{\ell}$ defined as the representative of the corresponding affine Weyl-group orbit described in the setup. PRV conjecture for the fusion product. Suppose $\lambda$ and $\mu$ are in $P_{\ell}$. Then, for any $w\in W_{\ell}$, the irreducible $\mathfrak{g}$-module $V(\overline{\lambda+w\mu}^{F})$ occurs with multiplicity at least one in the fusion product

$$V(\lambda)\otimes^{F}V(\mu).$$

This is the fusion-product analogue of the PRV conjecture, asserting that every affine-Weyl-group extremal candidate contributes an irreducible constituent. The supplied text gives no resolution status, so the conjecture is recorded as open.