Nontrivial units for Type 3 hyperelliptic curves over $\mathbb{F}_2$

Let $g,h\in\operatorname{\mathbb{F}}_2[x]$ define a Type 3 minimal representative with $h\ne 0$, so that $\deg g>\deg h$. The associated coordinate ring is

$$R=\operatorname{\mathbb{F}}_2[x,y]/(y^2+gy+h).$$

Nontrivial-unit conjecture. If $\deg g>\deg h$ and $h\ne 0$, then

$$R^\times\ne\{1\}.$$

This conjecture concerns the final untreated case of Type 3 minimal representatives; the preceding results handle the case $h=0$ and establish triviality in the case $g^2\mid h$. Its status is unresolved in the supplied source.