Asymptotic and catalytic containment of representations of $\mathsf{SU}(n)$

Given two finite-dimensional representations $\rho$ and $\sigma$ of $\mathsf{SU}(n)$, when is there $n \in \mathbb{N}$ such that $\rho^{\otimes n}$ is isomorphic to a subrepresentation of $\sigma^{\otimes n}$? When is there a third representation $\eta$ such that $\rho \otimes \eta$ is a subrepresentation of $\sigma \otimes \eta$? We call these the questions of asymptotic and catalytic containment, respectively. We answer both questions in terms of an explicit family of inequalities. These inequalities are almost necessary and sufficient in the following sense. If two representations satisfy all inequalities strictly, then asymptotic and catalytic containment follow (the former in generic cases). Conversely, if asymptotic or catalytic containment holds, then the inequalities must hold non-strictly. These results are an instance of a recent \emph{Vergleichsstellensatz} applied to the representation semiring.