Burton's worst-case vertex-normal-surface growth conjecture

Let $n$ be a positive integer other than $1,2,3,5$, and let $k$ be the integer determined by the congruence class of $n$ modulo $4$. A vertex normal surface is a vertex solution in the normal-surface solution space of a closed 3-manifold triangulation. Burton's worst-case growth conjecture. A tight upper bound on the number of vertex normal surfaces in a closed 3-manifold triangulation of size $n$ is

$$\begin{array}{ll} 17^k+k & \text{if } n=4k;\\ 581\cdot 17^{k-2}+k+1 & \text{if } n=4k+1;\\ 69\cdot 17^{k-1}+k & \text{if } n=4k+2;\\ 141\cdot 17^{k-1}+k+2 & \text{if } n=4k+3, \end{array}$$

and hence this upper bound grows asymptotically as $\Theta(17^{n/4})$. This is a conjectural sharp complexity bound for normal-surface enumeration in closed 3-manifold triangulations.