The quadratic-extension algebra-isomorphism conjecture for Hecke measure spaces

Let $F$ be the base field and let $E/F$ be a quadratic extension. For each division $F$-algebra $L$ of rank $n$, let $4\mathcal M^L_E4$ be the convolution algebra whose underlying vector space is canonically identified, independently of $L$, by the isomorphisms of Corollary 1.1.

Quadratic-extension algebra-isomorphism conjecture. For any quadratic extension $E/F$, these canonical isomorphisms are algebra isomorphisms.

The assertion extends the result known in the split case $E=F\oplus F$. The author expects that the Jacquet–Rallis relative trace formula could provide a proof.