Conjecture on the first Lyapunov exponent for strata with many poles

Fix a genus $g$. Let $Q_n$ be a sequence of connected components of strata, let $p_n$ be the number of poles in the stratum containing $Q_n$, and suppose that $p_n$ tends to infinity. Write $\lambda^+_1(Q_n)$ for the first positive Lyapunov exponent of $Q_n$. Many-poles Lyapunov-exponent conjecture. One has

$$\lambda^+_1(Q_n)=\mathcal O\left(1/\sqrt{p_n}\right).$$

The broader Grivaux–Hubert conjecture is refuted when the number of zeros stays bounded, but numerical experiments suggest this weaker statement when the number of zeros and poles both tend to infinity. Its status is not resolved in the supplied text.