The Clifford algebra isomorphism conjecture for geometric Waldspurger periods

Let $X$ be a smooth projective curve and let $E$ be an irreducible $\operatorname{SL}_2$-local system on $X$ such that, for every $d\ts 0$ and every covering $\phi:Y\to X$ in $\operatorname{RCov}^d$, one has $\operatorname{H}^0(Y,\phi^*E)=0$. Let $CL_E(X)\otimes CL_E^d$ be the corresponding sheaf of $\mathbb Z/2\mathbb Z$-graded Clifford algebras on $\operatorname{RCov}^d$, and let ${\cal F}_E$ be the associated local system, with $\operatorname{End}({\cal F}_E)\simeq {\cal F}_E^{\otimes 2}$. Clifford algebra isomorphism conjecture. Under these assumptions, the isomorphism $CL_E(X)\otimes CL_E^d\simeq \operatorname{End}({\cal F}_E)$ is an isomorphism of $\mathbb Z/2\mathbb Z$-graded sheaves of algebras on $\operatorname{RCov}^d$. This is a stronger compatibility statement identifying the Clifford algebra action with the full graded endomorphism algebra of the geometric Waldspurger local system; the source does not provide evidence resolving it.