Uniform mod-Poisson convergence for Abelian square-tiled surfaces

Let $\mathcal{H}_g$ be the Hodge bundle over $\mathcal{M}_g$, and let $p^{Ab}_g(k)$ be the probability that a random Abelian square-tiled surface in $\mathcal{H}_g$ has $k$ cylinders. Set

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

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

$$p^{Ab}_g(k+1)=e^{-\mu_g}\frac{\mu_g^k}{k!}\left(\frac{1}{\Gamma(t)}+o(1)\right).$$

The claim predicts that cylinder statistics for Abelian square-tiled surfaces are asymptotically modeled by a mod-Poisson law; the supplied text gives no resolution status beyond presenting it as a conjecture.