Jacquet–Langlands preservation conjecture for endo-classes

Let ${\rm G}={\rm GL}_m(D)$ and ${\rm H}={\rm GL}_{md}(F)$, where $D$ is a central division algebra over the non-archimedean local field $F$ of reduced degree $d$. Let $\mathcal{D}({\rm G})$ denote the set of isomorphism classes of essentially square-integrable representations of ${\rm G}$, and let $\boldsymbol\Theta_{\rm G}:\mathcal{D}({\rm G})\to\mathcal{E}(F)$ assign to each such representation its endo-class. Write ${\bf JL}$ for the Jacquet–Langlands correspondence from $\mathcal{D}({\rm G})$ to $\mathcal{D}({\rm H})$. Jacquet–Langlands endo-class conjecture. For any $\pi\in\mathcal{D}({\rm G})$, one has

$$\boldsymbol\Theta_{\rm H}({\bf JL}(\pi))=\boldsymbol\Theta_{\rm G}(\pi).$$

The conjecture proposes that Jacquet–Langlands transfer preserves the endo-class attached to an essentially square-integrable representation. It concerns the compatibility of the internal parametrization of representations of inner forms of general linear groups with transfer to the split form.