Optimal Gonality Conjecture for optimal curves

Let $C$ be an optimal curve over the finite field $\mathbb{F}_q$ of genus $g$, meaning that it has $N_q(g)$ rational points. The geometric gonality of a curve is the minimum degree of a morphism from its base change to $\mathbb{P}^1$ over the algebraic closure $\overline{\mathbb{F}}_q$.

Optimal Gonality Conjecture. Every optimal curve $C$ over $\mathbb{F}_q$ of genus $g$ has gonality at most $\operatorname{geomgon}$.

The conjecture predicts that optimal curves have especially low gonality. It holds for $q=2,3,4$ and $g\leq 5$, and also for $q=2$ and $g=6$; the source gives further examples in genera $7$, $8$, and $9$ over $\mathbb{F}_2$.