Uniform Abelian mod-Poisson convergence in strata

Let $\mathcal{C}$ be a non-hyperelliptic connected component of a stratum of Abelian differentials, and let $p_{\mathcal{C}}(k)$ be the probability that a random Abelian square-tiled surface in $\mathcal{C}$ has $k$ cylinders. Set

$$\mu_g=\log(4g-3).$$

Abelian-stratum mod-Poisson conjecture. For every $x>0$, uniformly for $k\in\{0,1,\ldots,\lfloor x\log(g)\rfloor\}$,

$$p^{Ab}_{\mathcal{C}}(k)=e^{-\mu_g}\frac{\mu_g^k}{k!}\left(\frac{1}{\Gamma(t)}+o(1)\right),$$

as $g\to\infty$, with the error term uniform over all non-hyperelliptic components of all strata of Abelian differentials. The claim is presented as a conjectural uniform extension of the preceding Abelian convergence statement; the supplied text gives no resolution status.