Wan conjectured that the variation of zeta functions along towers of curves associated to the $p$-adic etale cohomology of a fibration of smooth proper ordinary varieties should satisfy several stabilizing properties. The most basic of these conjectures state that the genera of the curves in these towers grow in a regular way. We state and prove a generalization of this conjecture, which applies to the graded pieces of the slope filtration of an overconvergent $F$-isocrystal. Along the way, we develop a theory of $F$-isocrystals with logarithmic decay and provide a new proof of the Drinfeld-Kedlaya theorem for curves.