Random mapping-torus volume ergodic and central-limit conjecture

Fix a generating set $\Gamma$ of the mapping class group of a surface of genus $g$. Let $\phi_N$ be a random word of length $N$, and let $V(\phi_N)$ be the volume of its mapping torus. Volume ergodic and central-limit conjecture. There is a constant $c(\Gamma)$ such that $V(\phi_N)/N$ converges almost surely to $c(\Gamma)$, and

$$\frac{1}{\sqrt{N}}\bigl(V(\phi_N)-Nc(\Gamma)\bigr)$$

converges in distribution to $N(0,\sigma^2)$ for some positive $\sigma=\sigma(\Gamma)$. The conjecture is motivated by empirical linear growth and approximately Gaussian volume distributions; both the almost-sure law and the central limit theorem remain open in this setting.