Strong uniform mod-Poisson convergence in quadratic-differential strata

Let $\mathcal{C}$ be a non-hyperelliptic connected component of a stratum of holomorphic quadratic differentials, and let $p_{\mathcal{C}}(k)$ be the probability that a random quadratic square-tiled surface in $\mathcal{C}$ has $k$ cylinders. Quadratic-stratum mod-Poisson conjecture. There exists a constant $R_2>1$ such that the mod-Poisson convergence holds with radius $R_2$ uniformly over all such components, namely

$$\sum_{k\geq1}p_{\mathcal{C}}(k)t^k =(\dim_{\mathbb{C}}\mathcal{C})^{\frac{t-1}{2}}\frac{\sqrt{\pi}}{\Gamma(t/2)}\left(1+O\left(\frac{1}{\dim_{\mathbb{C}}\mathcal{C}}\right)\right).$$

The error term is uniform over all non-hyperelliptic components of all strata of quadratic differentials and over $t$ in compact subsets of $|t|<R_2$. This is proposed as a strong uniform asymptotic statement, with no resolution supplied in the paper.