The monodromy of $F$-isocrystals with log-decay

Let U be a smooth geometrically connected affine curve over ${\mathbb{F}}_p$ with compactification X. Following Dwork and Katz, a $p$-adic representation $\rho$ of ${\pi }_1(U)$ corresponds to an 'etale $F$-isocrystal. By work of Tsuzuki and Crew an $F$-isocrystal is overconvergent precisely when $\rho$ has finite monodromy at each $x\in X-U$. However, in practice most F-isocrystals arising geometrically are not overconvergent and have logarithmic growth at singularities (e.g. characters of the Igusa tower over a modular curve). We give a Galois-theoretic interpretation of these log growth $F$-isocrystals in terms of asymptotic properties of higher ramification groups.