The generalized L-function rigidity conjecture for pointed curves

Let $(C,[M])$ and $(C',[M'])$ be smooth irreducible projective pointed curves over a finite field ${\mathbb F}_q$. Let ${\mathfrak m},{\mathfrak m}'$ be moduli on $C,C'$ disjoint from $M,M'$ respectively. Suppose the corresponding generalized Jacobians $J_{{\mathfrak m}}$ and $J_{{\mathfrak m}'}'$ have dimension at least two, and that there is a set-theoretic map

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

inducing an isomorphism of groups between $J_{{\mathfrak m}}({\mathbb F}_{q^n})$ and $J_{{\mathfrak m}'}'({\mathbb F}_{q^n})$ for every $n\geq 1$. The generalized L-function rigidity conjecture. If

$$L(T,C\otimes{\mathbb F}_{q^n},\chi)=L\bigl(T,C'\otimes{\mathbb F}_{q^n},\chi\circ\psi|_{J_{{\mathfrak m}'}'({\mathbb F}_{q^n})}\bigr)$$

for all $n$ and all characters $\chi$ of $J_{{\mathfrak m}}({\mathbb F}_{q^n})$, then $C$ and $C'$ are Frobenius twists of each other, and $\psi$ arises from a morphism of curves composed with a limit of Frobenius maps that sends $M$ to $M'$ and ${\mathfrak m}$ to ${\mathfrak m}'$. This is a rigidity statement recovering the pointed curve and the geometric correspondence from all finite-field extensions and character $L$-functions. The source states that the case ${\mathfrak m}=(0)$ was proved, whereas the general modulus case is presented as the conjectural extension.