Uniform square-root error bound for Abelian differential stratum volumes

Let $m=(m_1,\dots,m_n)$ be a partition indexed by $\Pi_g$, and let $\varepsilon_1(m)$ be the error term in the asymptotic volume formula for the stratum $\mathcal H_1(m)$.

Uniform error-bound conjecture. There exists a universal constant $C$ such that, for all $g$ and all $m\in\Pi_g$,

$$|\varepsilon_1(m)|\leq\frac{C}{\sqrt{g}}.$$

This is a quantitative strengthening of the preceding convergence assertion. The source further expects faster convergence for partitions obtained by adjoining entries equal to $1$, but does not state that expectation as a formal conjecture.