The asymptotic gonality-point equality conjecture for binary curves

Let $N_2(g,\gamma)$ denote the maximum number of $\b5\mathbb{F}_2$-rational points on a curve of genus $g$ and gonality $\gamma$. For a curve over $\mathbb{F}_2$ with a degree-$\gamma$ morphism to $\mathbb{P}^1$, one has $\#C(\mathbb{F}_2)\leq 3\gamma$.

Gonality-point equality conjecture. Fix $\gamma\geq 2$. For $g$ sufficiently large,

$$N_2(g,\gamma)=3\gamma.$$

The conjecture asserts that, after the low-genus obstructions disappear, the elementary gonality-point upper bound is attained. The supplied text does not indicate whether this has been resolved.