Recursive definability of curves and Jacobians in perfect fields

Let $K$ be a perfect field, let $g$ be a positive integer, and let $X$ and $Y$ be definable sets. A genus-$g$ curve over $K$ with Jacobian $J$ is represented by definable bijections between $X$ and $C(K)$ and between $Y$ and $J(K)$.

Recursive definability conjecture. There are formulas $\phi_{n,g}(\vec{x})$, $\psi_{n,g}(\vec{x},\vec{y})$, and $\rho_{n,g}(\vec{x},\vec{z})$, depending recursively on $n$ and $g$, such that the following are equivalent:

1. There is a genus-$g$ curve $C/K$ with Jacobian $J$ for which $X$ is in definable bijection with $C(K)$ and $Y$ is in definable bijection with $J(K)$.
2. There are $n\in\mathbb{N}$ and $\vec{a}\in\phi_{n,g}(K)$ such that $X$ is in definable bijection with $\psi_{n,g}(\vec{a},K)$ and $Y$ is in definable bijection with $\rho_{n,g}(\vec{a},K)$.

This is the underlying definability assertion used in the proposed proof of recursive axiomatizability. The surrounding discussion presents it as an informal formulation of what must be made precise, rather than as a resolved result.