Davydov–Marcugini–Pambianco orbit-count conjecture for generic lines in $PG(3,q)$

Let $q$ be a power of $3$, let $G=PGL_2(q)$ act on the lines of $PG(3,q)$, and let $C$ be the twisted cubic. A line is generic if it neither intersects $C$ nor lies in any osculating plane of $C$; write $\mathcal{O}_6$ for the class of generic lines. Davydov–Marcugini–Pambianco's conjecture. The number of $G$-orbits of generic lines of $PG(3,q)$ is $2q-3$, of which there are $q/3$, $q-1$, and $(2q-6)/3$ orbits of sizes $|G|$, $|G|/2$, and $|G|/4$, respectively. This conjecture gives the proposed classification of the generic-line class $\mathcal{O}_6$ into $G$-orbits in characteristic $3$, a case that had remained open after the corresponding problems in characteristics different from $2$ and $3$, and in characteristic $2$, were solved.