Distribution of point classes in Picard-group cosets for odd-degree hyperelliptic curves

Let $K$) be the function field of a hyperelliptic curve $X$ of odd degree. Write

$$\operatorname{Pic} K/2\operatorname{Pic} K=\operatorname{Pic}^0 K/2\operatorname{Pic}^0 K\oplus\mathbb{Z}_2.$$

For a point $\mathfrak{p}\in X$, let $\deg\mathfrak{p}$ denote its degree and let its class in this quotient be understood via this decomposition. Distribution conjecture. There exists an integer $N_0\in\mathbb{N}$ such that, for every $\mathfrak{D}\in\operatorname{Pic}^0 K/2\operatorname{Pic}^0 K$ and every integer $n>N_0$, there is a point $\mathfrak{p}\in X$ with $\deg\mathfrak{p}=n$ whose class in $\operatorname{Pic} K/2\operatorname{Pic} K$ is $(\mathfrak{D},\deg\mathfrak{p}\bmod 2)$. Thus, from some degree onward, every coset contains the class of a point of each prescribed degree, subject only to the parity condition forced by the degree map.