  We discuss a test of a recently proposed approach to determine average
pairing matrix elements within a given interval of single-particle states (sp)
around the Fermi level $\lambda$ as obtained in the so-called uniform gap
method (UGM). It takes stock of the crucial role played by the averaged sp
level density $\tilde{\rho}(e)$. These matrix elements are deduced within the
UGM approach, from microscopically calculated $\tilde{\rho}(e)$ and gaps
obtained from analytical formulae of a semi-classical nature. Two effects
generally ignored in similar fits have been taken care of. They are: (a) the
correction for a systematic bias in choosing to fit pairing gaps corresponding
to equilibrium deformation solutions as discussed by M\"{o}ller and Nix [Nucl.
Phys. A 476, 1 (1992)] and (b) the correction for a systematic spurious
enhancement of $\tilde{\rho}(e)$ for protons in the vicinity of $\lambda$,
because of the local Slater approximation used for the treatment of the Coulomb
exchange terms in most calculations (see e.g. [Phys. Rev C 84, 014310 (2011)]).
This approach has been deemed to be very efficient upon performing Hartree-Fock
+ BCS (with seniority force and self-consistent blocking when dealing with odd
nuclei) calculations of a large sample of well and rigidly deformed even-even
rare-earth nuclei. The reproduction of their experimental moments of inertia
has been found to be at least of the same quality as what has been obtained in
a direct fit of these data [Phys. Rev C 99, 064306 (2019)]. We extend here the
test of our approach to the reproduction, in the same region, of three-point
odd-even mass differences centered on odd-$N$ or odd-$Z$ nuclei. The agreement
with the data is again roughly of the same quality as what has been obtained in
a direct fit, as performed in [Phys. Rev C 99, 064306 (2019)].
