Tight upper-bound conjecture for admissible vertex counts of normal surfaces

Let $n$ be the number of tetrahedra in a triangulation, and let $\sigma$ denote its admissible vertex count. For positive $n\neq 1,2,3,5$, equation~ gives the proposed upper bound.

Tight upper-bound conjecture. For every positive $n\neq 1,2,3,5$, equation~ gives a tight upper bound on the admissible vertex count $\sigma$. Consequently,

$$\sigma\in O(17^{n/4}).$$

The conjecture is motivated by the census, in which the pathological triangulations give the maximum observed admissible vertex counts wherever they are defined. It predicts the observed maximum growth rate, but the supplied text gives no resolution beyond these computational observations.