The strong separability conjecture for the tensor-product subgroup

Let $G=SL_{n^2}(\mathbb{C})$ and let $H=SL_n(\mathbb{C})\times SL_n(\mathbb{C})$ have its natural embedding in $SL(\mathbb{C}^n\otimes\mathbb{C}^n)$. A subgroup is strongly $G$-separable if every relevant nontrivial irreducible $H$-module occurs in infinitely many irreducible non-$H$-admissible $G$-modules. Strong tensor-product separability conjecture. The subgroup

$$SL_n(\mathbb{C})\times SL_n(\mathbb{C})$$

is strongly separable in $SL_{n^2}(\mathbb{C})$. Equivalently, for every nontrivial Weyl module $V_\lambda(n)\otimes V_\mu(n)$ with $|\lambda|\equiv|\mu|\pmod n$, there exist infinitely many Weyl modules $V_\rho(n^2)$ whose restriction contains $V_\lambda(n)\otimes V_\mu(n)$ but not the trivial module. This strengthens ordinary separability and is reformulated in the paper through symmetric-group tensor-product multiplicities.