The group-theoretic Torelli conjecture for curves over finite fields

Let $k=\overline{\mathbb F}_p$ with $p\geq 3$, and let $C$ and $\tilde{C}$ be smooth projective curves over $k$ of genus at least $2$, with Jacobians $J$ and $\tilde{J}$. Suppose that $(C,J)$ and $(\tilde{C},\tilde{J})$ are related by an isomorphism of pairs as in Theorem~. The group-theoretic Torelli conjecture. Under the assumptions of Theorem~, $C$ and $\tilde{C}$ are isomorphic as algebraic varieties, modulo Frobenius twisting. The theorem in the surrounding text proves only that $J$ and $\tilde{J}$ are isogenous; the asserted recovery of the curve itself is a stronger Torelli-type statement, and its resolution is not specified here.