Fibonacci-growth conjecture for mean admissible vertex counts

For each $n$, let $\overline{\sigma}_{n}$ denote the mean admissible vertex count among all triangulations of size $n$, and let

$$\phi=\frac{1+\sqrt{5}}{2}.$$

Fibonacci-growth conjecture. For every $n\geq 3$, the mean admissible vertex count satisfies

$$\overline{\sigma}_{n}<\overline{\sigma}_{n-1}+\overline{\sigma}_{n-2}.$$

Consequently, $\overline{\sigma}_{n}$ is bounded above by $O(\phi^n)$.

The conjecture is suggested by the census, where the stated inequality holds throughout the tested range and the average growth lies below the Fibonacci growth rate. The supplied text gives no proof or resolution.