Aubert–Zelevinsky duality conjecture for unipotent representations

Let $\textmd{k}$ be a nonarchimedean local field of characteristic $0$, let $\textmd{G}$ be a reductive algebraic group over $\textmd{k}$, and let $\{{\it e},{\it h},{\it f}\}$ be an $\mathfrak{sl}_2$-triple with ${\it h}\in\mathfrak{t}^{\vee}_r$. Let $\mathcal{S}\subset\mathfrak{g}^{\vee}(2)$ be an orbit and let $(\mathcal{S}_{Ft},\mathcal{L}_{Ft})$ be the orbit and local system obtained by Fourier transform. Write $X(q^{\frac{1}{2}{\it h}},\mathcal{S},\mathcal{L})$ for the corresponding irreducible unipotent representation, and let $AZ$ denote Aubert–Zelevinsky duality.

Aubert–Zelevinsky duality conjecture. If $X=X(q^{\frac{1}{2}{\it h}},\mathcal{S},\mathcal{L})$ is an irreducible unipotent representation in $\Pi_{q^{\frac{1}{2}{\it h}}}^{\text{Lus}}(\textmd{G}(\textmd{k}))$, then

$$AZ(X)=X(q^{\frac{1}{2}{\it h}},{}^t\mathcal{S}_{Ft},{}^t\mathcal{L}_{Ft}).$$

The conjecture is known for all representations with Iwahori-fixed vectors, but the source gives no general resolution.