The asymptotic second Lyapunov exponent conjecture for strata of holomorphic differentials

Let ${\cal M}^{\cal H}_{(2g-2)}$ and ${\cal M}^{\cal H}_{(g-1,g-1)}$ denote the hyperelliptic components of the corresponding moduli spaces of holomorphic differentials, and let $\lambda_2$ be the second Lyapunov exponent. The asymptotic second Lyapunov exponent conjecture. For the hyperelliptic components,

$$\lim_{g\to\infty}\lambda_2=1.$$

For all other components and other moduli spaces of holomorphic differentials,

$$\lim_{g\to\infty}\lambda_2=\frac12.$$

The conjecture concerns the asymptotic deviation from the average in interval-exchange and related dynamical systems. The source gives numerical motivation but no proof or resolution.