Few-zeros conjecture for ample line bundles on curves

Let $K$ be an algebraically closed field other than $\overline{\mathbb F}_p$, and let $C$ be a smooth projective curve over $K$.

Few-zeros conjecture. For “most” ample line bundles $L$, every section of $L^m$ has at least $g(C)$ zeros for every $m\geq 1$.

The statement is known when $K$ is uncountable, while the most interesting open case mentioned is $K=\overline{\mathbb Q}$. The paper also proves the nodal rational curve cases and gives examples showing that the bound can be nontrivial.