The conjecture on Aubert-Zelevinsky involutions of ABV-packets

Let $G$ be a reductive group over a non-Archimedean field, let $\phi$ be an $L$-parameter, and let $\Pi_{\phi}^{\mathrm{ABV}}$ denote its ABV-packet. Let $\widehat{\phi}$ be the Pyasetskii involution of $\phi$, and let $\widehat{\pi}$ denote the Aubert-Zelevinsky involution of a representation $\pi$. ABV-packet involution conjecture. The Aubert-Zelevinsky involution sends the ABV-packet of $\phi$ to the ABV-packet of $\widehat{\phi}$:

$$\Pi_{\widehat{\phi}}^{\mathrm{ABV}}=\\{\widehat{\pi}\mid \pi\in\Pi_{\phi}^{\mathrm{ABV}}\\}.$$

This is proposed as an extension of the known compatibility for local Arthur packets; the statement is presented as an expectation for ABV-packets beyond Arthur type.