The tensor-product embedding conjecture for the dominance order on pairs

Let $P^+$ be the set of dominant integral weights, and let $V(\gamma)$ denote the irreducible highest weight representation of highest weight $\gamma$. For pairs of dominant weights with $\lambda+\mu=\lambda'+\mu'$, define $(\lambda,\mu)\preceq(\lambda',\mu')$ when

$$\min\{\lambda(\beta^\vee),\mu(\beta^\vee)\}\leq\min\{\lambda'(\beta^\vee),\mu'(\beta^\vee)\}$$

for every positive coroot $\beta^\vee$. The tensor-product embedding conjecture. If $(\lambda,\mu)\preceq(\lambda',\mu')$, then

$$V(\lambda)\otimes V(\mu)\subseteq V(\lambda')\otimes V(\mu').$$

The conjecture connects the order on pairs of dominant weights with embeddings of tensor products and is motivated by fusion products of cyclic current-algebra modules. Its resolution is not supplied in the source.