Isospectrality determines the zeta functions of canonical models

Let $tilde{G}$ be the group occurring in the Shimura-variety construction, let $tilde{K}_1,tilde{K}_2$ be compact open subgroups of $tilde{G}(F)$, and let $\Gamma_{K_1},\Gamma_{K_2}$ be the associated lattices. Assume that these lattices are torsion-free modulo their centers. Let $M_{K_1}$ and $M_{K_2}$ be the associated spaces, and let $M_{tilde{K}_1}$ and $M_{tilde{K}_2}$ be their corresponding non-connected Shimura varieties with canonical models.

Canonical-model zeta conjecture. If $M_{K_1}$ and $M_{K_2}$ are isospectral, then the Hasse–Weil zeta functions of the canonical models of $M_{tilde{K}_1}$ and $M_{tilde{K}_2}$ are equal.

The conjecture refines the earlier question for connected arithmetic curves by using the better-behaved non-connected Shimura varieties associated with inner forms of $GL(2)$. The supplied text gives no resolution, so the conjecture remains open.