Aubert dual conjecture for representations parametrized by extended multi-segments

Let $\mathcal{E} = \cup_{\rho}\{ ([A_i,B_i]_{\rho}, l_i, \eta_i) \}_{i \in (I_\rho,>)}$ be an extended multi-segment, and let

$$\hat{\mathcal{E}} = \cup_{\rho}\{ ([A_i,-B_i]_{\rho}, \hat{l}_i, \hat{\eta}_i) \}_{i \in (I_\rho,\hat{>})}$$

be the extended multi-segment defined from $\mathcal{E}$. Write $\pi(\mathcal{E})$ for the representation associated with $\mathcal{E}$ and $\hat{\pi}(\mathcal{E})$ for its Aubert dual. Aubert dual conjecture. If $\pi(\mathcal{E}) \not= 0$, then

$$\hat{\pi}(\mathcal{E}) \cong \pi(\hat{\mathcal{E}}).$$

The conjecture proposes an explicit description of the Aubert involution on the representations constructed from extended multi-segments. Aubert duality is known to exchange the packets attached to an $A$-parameter and its parameter obtained by exchanging the two $\mathrm{SL}_2(\mathbb{C})$ factors, but the claimed formula at the level of extended multi-segments is presented here as a conjecture.