Universality conjecture for curves over finite algebraic closures

Let $C$ be a smooth complete curve of genus $g(C)\ge 2$ defined over $\overline{{\mathbf F}}_p$, where $p\ge 2$. A curve is universal if every smooth projective curve admits an unramified cover that maps surjectively onto it.

Universality conjecture. Every such curve $C$ is universal.

The preceding result proves this for hyperelliptic curves in characteristic $p\ge 5$, while the statement proposes universality for all smooth complete curves of genus at least two over $\overline{{\mathbf F}}_p$.