Bernstein's unitarizability conjecture for the Aubert–Zelevinsky duality of $G_2$

Let $G_2$ be the reductive group under consideration, and let $\pi$ be an irreducible unitarizable representation of $G_2$. The Aubert–Zelevinsky duality of $\pi$ is denoted by $\operatorname{D}(\pi)$. Bernstein's conjecture. If $\pi$ is an irreducible unitarizable representation of $G_2$, then its Aubert–Zelevinsky duality $\operatorname{D}(\pi)$ is also unitarizable. The paper verifies many cases of this conjecture using computations for $G_2$ and earlier work, but the statement is not established in full here.