Refined Eskin–Zorich asymptotics for volumes and Siegel–Veech constants

Let $\mu$ index a stratum of Abelian differentials, let ${\rm \dim}({\mathcal H}(\mu))$ denote its dimension, and let $\epsilon_1(\mu)$ and $\epsilon_2(\mu)$ be the error terms in the volume and area Siegel–Veech constant asymptotics. Refined Eskin–Zorich conjecture. The functions satisfy

$$-\epsilon_1(\mu)=\frac{\pi^2}{6\,{\rm \dim}({\mathcal H}(\mu))}(1+\epsilon_1'(\mu)),$$ $$-\epsilon_2(\mu)=\frac{1}{2\,{\rm \dim}({\mathcal H}(\mu))}(1+\epsilon_2'(\mu)),$$

where $\epsilon_1'(\mu)$ and $\epsilon_2'(\mu)$ tend uniformly to $0$ as $g$ tends to infinity. This conjecture refines the leading-order uniform asymptotics by predicting the first correction in terms of the stratum dimension. The source presents it as a proposed straightening of the Eskin–Zorich conjectures and gives numerical and special-stratum evidence, but no general proof.