Grivaux–Hubert conjecture on Lyapunov exponents with many poles

A translation surface is a surface equipped with a flat structure whose singularities may include poles, and its positive Lyapunov exponents are the exponents of the Kontsevich–Zorich cocycle that are greater than zero. Let $p$ denote the number of poles. Grivaux–Hubert conjecture. The positive Lyapunov exponents associated to a translation surface have a uniform bound depending only on $p$. This bound tends to zero as $p$ tends to infinity. The conjecture was motivated by the observation that poles slow down the linear flow. The source later explains that the claim is false in this generality when the number of zeros remains bounded, because genus-one covers can have arbitrarily many poles while their first Lyapunov exponent remains $2/3$.