  In this paper we continue the program pioneered by D'Hoker and Phong, and
recently advanced by Cacciatori, Dalla Piazza, and van Geemen, of finding the
chiral superstring measure by constructing modular forms satisfying certain
factorization constraints. We give new expressions for their proposed ans\"atze
in genera 2 and 3, respectively, which admit a straightforward generalization.
We then propose an ansatz in genus 4 and verify that it satisfies the
factorization constraints and gives a vanishing cosmological constant. We
further conjecture a possible formula for the superstring amplitudes in any
genus, subject to the condition that certain modular forms admit holomorphic
roots.
