Kontsevich–Zorich conjecture on the asymptotic second Lyapunov exponent

Let $g$ denote the genus, let $\theta_2$ be the second Lyapunov exponent of the Kontsevich–Zorich cocycle over the indicated moduli space of translation surfaces, and let $\mathcal{M}^{\mathcal{H}}_{2g-2}$ and $\mathcal{M}^{\mathcal{H}}_{(g-1,g-1)}$ denote the hyperelliptic components. Consider also the other connected components and moduli spaces of holomorphic differentials.

Kontsevich–Zorich conjecture. For the hyperelliptic components $\mathcal{M}^{\mathcal{H}}_{2g-2}$ and $\mathcal{M}^{\mathcal{H}}_{(g-1,g-1)}$,

$$\lim_{g\to\infty}\theta_2=1.$$

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

$$\lim_{g\to\infty}\theta_2=\frac{1}{2}.$$

The paper presents this as a conjecture concerning the large-genus limit of the second exponent and studies bounds related to it using invariant submanifolds and a twisted cocycle. The supplied text gives no evidence of resolution.