The generalized Jacobian reconstruction conjecture for pointed curves

Let $(C,[M])$ and $(C',[M'])$ be smooth, projective, irreducible pointed curves over ${\mathbb F}_q$, with moduli ${\mathfrak m},{\mathfrak m}'$ on $C,C'$ respectively. Suppose the generalized Jacobians $J_{{\mathfrak m}}$ and $J_{{\mathfrak m}'}'$ have dimension at least two. Let $U$ and $U'$ be the complements of the supports of ${\mathfrak m}$ and ${\mathfrak m}'$ in $C$ and $C'$, respectively. If

$$\psi:J_{{\mathfrak m}}(\overline{{\mathbb F}}_q)\to J_{{\mathfrak m}'}'(\overline{{\mathbb F}}_q)$$

is a group isomorphism satisfying $\psi(U(\overline{{\mathbb F}}_q))=U'(\overline{{\mathbb F}}_q)$ with respect to the Abel–Jacobi embeddings determined by $M$ and $M'$, then $\psi$ arises from a morphism of curves composed with a limit of Frobenius maps sending ${\mathfrak m}$ to ${\mathfrak m}'$. The generalized Jacobian reconstruction conjecture. More precisely, there exists an integer $m$, an isomorphism

$$\alpha:\operatorname{Frob}^m(C)\simeq C',$$

and a generalized Frobenius $\beta:\overline{{\mathbb F}}_q\to\overline{{\mathbb F}}_q$ restricting to $\operatorname{Frob}^{-m}$ on ${\mathbb F}_q$, such that $\alpha\circ\beta:C\to C'$ sends ${\mathfrak m}$ to ${\mathfrak m}'$ and $M$ to $M'$, and induces $\psi$ on generalized Jacobians. This strengthens the known zero-modulus reconstruction result to generalized Jacobians with moduli.