Local Gan–Gross–Prasad conjecture for Vogan packets

Let $G_n^*$ and $H_m^*$ be a relevant pair of $F$-quasisplit classical groups, let $R_{{\mathcal {O}}_\ell}(F)=H_m(F)\ltimes V_{\underline{p}_\ell}(F)$ be the Bessel subgroup attached to the relevant orbit, and let $\psi_{{\mathcal {O}}_\ell}$ be its Bessel character. For a local $L$-parameter $\phi\in\widetilde{\Phi}^{+}_{\mathrm{unit}}(G_n^*\times H_m^*)$, the local Gan–Gross–Prasad conjecture for Vogan packets.

$$\sum_{\pi\otimes\sigma\in\widetilde{\Pi}_\phi[G_n^*\times H_m^*]}\dim\operatorname{Hom}_{R_{{\mathcal {O}}_\ell}(F)}(\pi\otimes\sigma,\psi_{{\mathcal {O}}_\ell})=1.$$

This strengthens the multiplicity-one statement for local Bessel models from individual representations to an entire Vogan packet. It is open over archimedean local fields for orthogonal groups.