Ben-Zvi–Sakellaridis–Venkatesh conjecture on duals of BZSV quadruples

Let $\Delta=(G,H,\iota,\rho_H)$ be a BZSV quadruple, let $L$ be the associated Levi subgroup, and set $\Delta_{\mathrm{red}}=(L,H,1,\rho_{H,\iota})$. Suppose the dual of $\Delta_{\mathrm{red}}$ is

$$\hat{\Delta}_{\mathrm{red}}=(\hat{L},\hat{H}_L',\hat{\iota}',\rho').$$

For a representation $\rho$ of a Levi subgroup $L$ of $G$, write $(\rho)_L^G$ for the representation obtained by transporting the highest weights of its irreducible constituents to dominant highest weights of $G$.

Ben-Zvi–Sakellaridis–Venkatesh duality conjecture. The dual of $\Delta$ is

$$(\hat{G},\hat{H}',\hat{\iota}',(\rho')_{\hat{H}_L'}^{\hat{H}'}),$$

where $\hat{H}_L'$ is a Levi subgroup of $\hat{H}'$ and $\hat{H}'$ is generated by $\hat{H}_L'$ and

$$\{\operatorname{Im}(\iota_\alpha)\mid \alpha\in\Delta_{\hat{G}}-\Delta_{\hat{M}}\}.$$

Here $\Delta_{\hat{G}}$ and $\Delta_{\hat{M}}$ are the sets of simple roots of $\hat{G}$ and $\hat{L}$, respectively, and $\iota_\alpha:\operatorname{SL}_2\to\hat{G}$ is the embedding associated with $\alpha$.

This conjecture describes how the dual of a general BZSV quadruple should be reconstructed from the dual of its reductive reduction. The source attributes it to Ben-Zvi, Sakellaridis, and Venkatesh and gives no evidence of a resolution.