  We report calculations of the superfluid pairing gap in neutron matter for
the $^1S_0$ components of the Reid soft-core $V_6$ and the Argonne $V_{4}'$
two-nucleon interactions. Ground-state calculations have been carried out using
the central part of the operator-basis representation of these interactions to
determine optimal Jastrow-Feenberg correlations and corresponding effective
pairing interactions within the correlated-basis formalism (CBF), the required
matrix elements in the correlated basis being evaluated by Fermi
hypernetted-chain techniques. Different implementations of the
Fermi-Hypernetted Chain Euler-Lagrange method (FHNC-EL) agree at the percent
level up to nuclear matter saturation density. For the assumed interactions,
which are realistic within the low density range involved in $^1S_0$ neutron
pairing, we did not find a dimerization instability arising from divergence of
the in-medium scattering length, as was reported recently for simple
square-well and Lennard-Jones potential models (Phys. Rev. A {\bf 92}, 023640
(2015)).
