Asymptotic eigenvalue conjecture for random mapping tori

Let $N$ denote the length of a random monodromy word and let $N^3$ be the associated mapping torus. Write $\lambda_1(N)$ for the bottom eigenvalue of the Laplacian. Asymptotic eigenvalue conjecture. With the notation of the paper,

$$\lambda_1(N)\sim \frac{C}{N}.$$

The preceding corollary gives only two-sided bounds of orders $N^{-2}$ and $N^{-1}$; the conjecture predicts the sharper asymptotic order and remains open.