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