Burton's average vertex-normal-surface growth conjecture

For each $n$, let $\overline{\sigma}_n$ denote the average number of vertex normal surfaces over all closed 3-manifold triangulations of size $n$, counted up to isomorphism. A vertex normal surface is a vertex solution in the normal-surface solution space. Burton's average-growth conjecture. For every $n\geq 3$,

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

and consequently

$$\overline{\sigma}_n\in O\left(\left[\frac{1+\sqrt{5}}{2}\right]^n\right).$$

This predicts Fibonacci-type control of the average complexity of vertex-normal-surface enumeration.