Multiplicity-one conjecture for the Σ × σ period

Let $H=\operatorname{GL}(2,F)\times_{F^\times}\operatorname{GL}(2,F)$, let $\pi$ and $\sigma$ be as in the preceding uniqueness setting, and let $s_0$ be the relevant complex parameter. Let $\Sigma$ be the reducible principal-series representation of $\operatorname{GL}(2,F)$ having the Steinberg representation as subrepresentation.

Σ × σ multiplicity-one conjecture. With these notations,

$$\dim\operatorname{Hom}_H(\pi\otimes(\Sigma\boxtimes\sigma),\mathbf C)=1,$$

whether or not $s_0$ is exceptional.

This is a formulation of the desired uniqueness statement for the period space; the source does not establish it in general, so its status remains open.