  The contribution from even spin structures to the genus-two amplitude for
five massless external NS states in Type II and Heterotic superstrings is
evaluated from first principles in the RNS formulation. Using chiral splitting
with the help of loop momenta this problem reduces to the evaluation of the
corresponding chiral amplitude, which is carried out using the same techniques
that were used for the genus-two amplitude with four external NS states. The
results agree with the parity-even NS components of a construction using chiral
splitting and pure spinors given in earlier companion papers arXiv:2006.05270
and arXiv:2008.08687.
