Isolated representations are of critical type

Let $\pi$ be an isolated representation in the unitary dual of a classical group. A representation is of critical type if it is an irreducible subquotient of $\rho_1\times\dots\times\rho_k\rtimes\sigma$, where $\sigma$ is an irreducible cuspidal representation of a classical group and, for each $i$, $\rho_i^u$ is self-dual, the set $\{e(\rho_j):\rho_j^u\cong\rho_i^u\}$ is a possibly multiplicity-bearing $\mathbb Z$-segment in $\frac12\mathbb Z$, and that segment contains the reducibility exponent $\alpha_{\rho_i^u,\sigma}$. Critical-type conjecture. The representation $\pi$ is of critical type. The conjecture is known for unramified representations, while the general assertion remains open.