  We demonstrate (under a modest assumption) that the sums over spin-structures
of the simplest combinations of fermionic correlators (Szego kernels) and
DHP/CDG/Grushevsky NSR measures vanish at least on the hyperelliptic loci in
the moduli space of Riemann surfaces -- despite the violation of the theta_e^4
hypothesis at g>2. This provides an additional important support to validity of
these measures and is also a step towards a proof of the non-renormalization
theorems in the NSR approach.
