Uniformity conjecture for period transfer when n is even

Let $E/F$ be the quadratic extension, let $G'=\operatorname{GL}_{2n}$, and let $\pi'$ be a cuspidal representation of $G'(\mathbf A)$. For each $D\in X(E:F:\pi')$, let $\pi_D$ be the inverse Jacquet–Langlands transfer to the corresponding inner form $G_D(\mathbf A)$, and let $H$ be the associated subgroup. Uniformity conjecture. If $n$ is even, then for any $D_1,D_2\in X(E:F:\pi')$,

$$\pi_{D_1}\text{ is }H\text{-distinguished}\quad\Longleftrightarrow\quad\pi_{D_2}\text{ is }H\text{-distinguished}.$$

This is the proposed global “anti-dichotomy” principle: for even $n$, distinction should be uniform across all inner forms for which the inverse transfer exists. The source presents it as a conjecture motivated by the relative trace formula and explicitly distinguishes it from, and notes that it is not implied by, the local statement.