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Let $\pi:X\to Y$ be an $n$-dimensional iterated Kodaira fibration with fiberof genus $g$ and injective monodromy. Llosa Isenrich and Py proved that we canpass to a finite index subgroup of $\pi_1(X)$ to get the base space of ann+1-dimensional iterated Kodaira fibration with injective monodromy and theyasked about bounding the index of such a group. We provide a bound on thisindex.

Bounds on degrees of covers with injective monodromy in iterated Kodaira Fibrations