Asymptotic count conjecture for trihexes

Let $v$ be the number of vertices of a trihex, and let $\alpha(v)$ and $\beta(v)$ denote respectively the number of equivalence classes and the number of graph isomorphism classes of trihexes with $v$ vertices. Let $\sigma(v)$ be the number of signatures representing trihexes with $v$ vertices.

Asymptotic count conjecture. As $v \to \infty$, the counts $\alpha(v)$ and $\beta(v)$ are respectively asymptotic to

$$\dfrac{1}{3}\sigma(v) \quad\text{and}\quad \dfrac{1}{6}\sigma(v).$$

The bounds preceding the conjecture arise because each trihex can have up to three signatures up to equivalence and up to six signatures up to graph isomorphism. Computations suggest that the two counts are close to the corresponding lower bounds, but the asymptotic assertion remains unproved in the supplied text.