The rational basis-function conjecture for non-hyperelliptic Jacobians

Let $\mathcal{C}$ be a non-hyperelliptic curve, let $\operatorname{Jac}(\mathcal{C})$ be its Jacobian, let $\Sigma$ denote the relevant theta divisor, and let the basis $\wp$-functions be the functions introduced for the Weierstrass gap sequence of $\mathcal{C}$. Rational basis-function conjecture. The abelian function field $\mathfrak{A}(\mathcal{C})$ associated with $\mathcal{C}$ is a ring of rational functions in the basis $\wp$-functions. Equivalently, every meromorphic function on $\operatorname{Jac}(\mathcal{C}) \backslash \Sigma$ is represented as a rational function in the basis $\wp$-functions. This extends the preceding polynomial-ring description established in the hyperelliptic case to non-hyperelliptic curves; the supplied text does not state whether the claim has been proved or remains open.