Weak separation characterizes simplicity of tensor products of fundamental modules

Let $J,J'$ be $k$-element subsets of $[n]$, and let $L(M_J)$ and $L(M_{J'})$ be the corresponding simple $U_q(\widehat{\mathfrak{sl}_k})$-modules. Weak-separation tensor-product conjecture. If

$$L(M_J)\otimes L(M_{J'})$$

is simple, then $J$ and $J'$ are weakly separated. Together with the proved converse, this conjecture would characterize simplicity of these tensor products in terms of weak separation; the converse is established in the paper, while this implication is left conjectural.