The saturation conjecture for tensor products of unitary-group representations

Let $\lambda$, $\mu$, and $u$ be weakly decreasing sequences of $n$ integers. Let $\lambda\boxplus\mu\sim_q\nu$ denote the corresponding quantum relation, and let $\lambda\boxplus\mu\sim_c\nu$ denote the classical relation defined by spectra of sums of Hermitian matrices. The theorem in the source states that the classical relation \implies the quantum relation after multiplying all three sequences by some positive integer $N$.

Saturation conjecture. One can take $N=1$ in the above theorem. Equivalently, for integer $\lambda$, $\mu$, and $u$, the relations $\lambda\boxplus\mu\sim_c\nu$ and $\lambda\boxplus\mu\sim_q\nu$ are equivalent.

This asserts that the asymptotic quantum condition has no nontrivial scaling obstruction. The source supplies no resolution status for this conjecture.