  Primordial black holes (PBH), supposedly formed in the very early Universe,
have been proposed as a possible viable dark matter candidate. In this work we
characterize the expected gravitational wave (GW) losses from a population of
PBHs orbiting Sgr A$^{\star}$, the super-massive black hole at the Galactic
center (GC), and assess the signal detectability by the planned space-borne
interferometer LISA and by the proposed next generation space-borne
interferometer $\mu$Ares. Assuming that PBHs indeed form the entire diffuse
mass allowed to reside within the orbit of the S2 star, we compute an upper
limit to the expected GW signal both from resolved and non-resolved sources,
under the further assumptions of monochromatic mass function and thermally
distributed eccentricities. By comparing with our previous work where PBHs on
circular orbits were assumed, we show for 1 M$_{\odot}$ PBHs how the GW signal
from high harmonics over a 10 year data stream increases by a factor of six the
chances of LISA detectability, from the $\approx 10\%$ of the circular case, to
$\approx 60\%$, whereas multiple sources can be identified in $20\%$ of our
mock populations. The background signal, made by summing up all non resolved
sources, should be certainly detectable thanks to the PBHs with higher
eccentricity evolving under two body relaxation. In the case of $\mu$Ares,
because of its improved sensitivity in the $\mu$Hz band, one third of the
entire population of PBHs orbiting Sgr A$^{\star}$ would be resolved. The
background noise from the remaining non resolved sources should be detectable
as well. Finally we present the results for different PBH masses.
