  There is a long-standing belief that the modular tensor categories
$\mathcal{C}(\mathfrak{g},k)$, for $k\in\mathbb{Z}_{\geq1}$ and
finite-dimensional simple complex Lie algebras $\mathfrak{g}$, contain
exceptional connected \'etale algebras at only finitely many levels $k$. This
premise has known implications for the study of relations in the Witt group of
nondegenerate braided fusion categories, modular invariants of conformal field
theories, and the classification of subfactors in the theory of von Neumann
algebras. Here we confirm this conjecture when $\mathfrak{g}$ has rank 2,
contributing proofs and explicit bounds when $\mathfrak{g}$ is of type $B_2$ or
$G_2$, adding to the previously known positive results for types $A_1$ and
$A_2$.
