Distinction converse conjecture for supercuspidal representations

Let $E/{\mathcal F}$ be the quadratic extension implicit in the distinction notation, with $n\ge3$. Let $\pi$ be a supercuspidal representation of $\operatorname{GL}_n(E)$, and for $1\le r\le\left[\frac n2\right]$ let $\rho$ range over irreducible generic $\operatorname{GL}_r({\mathcal F})$-distinguished representations of $\operatorname{GL}_r(E)$. Distinction converse conjecture. The representation $\pi$ is $\operatorname{GL}_n({\mathcal F})$-distinguished if

$$\gamma\left(\frac12,\pi\times\rho,\psi_E\right)=1$$

for every such $\rho$. This extends the known converse results using twists through rank $\left[\frac n2\right]$; the general assertion is presented as conjectural.