  In a recent work, Baldi et al. highlighted the issue of cosmic degeneracies,
consisting in the fact that the standard statistics of the large-scale
structure might not be sufficient to conclusively test cosmological models
beyond $\Lambda $CDM when multiple extensions of the standard scenario coexist
in nature. In particular, it was shown that the characteristic features of an
$f(R)$ Modified Gravity theory and of massive neutrinos with an appreciable
total mass $\Sigma _{i}m_{\nu _{i}}$ are suppressed in most of the basic
large-scale structure observables for a specific combination of the main
parameters of the two non-standard models. In the present work, we explore the
possibility that the mean specific size of the supercluster spines -- which was
recently proposed as a non-standard statistics by Shim and Lee to probe gravity
at large scales -- can help to break this cosmic degeneracy. By analyzing the
halo samples from N-body simulations featuring various combinations of $f(R)$
and $\Sigma _{i}m_{\nu _{i}}$ we find that -- at the present epoch -- the value
of $\Sigma _{i}m_{\nu _{i}}$ required to maximally suppress the effects of
$f(R)$ gravity on the specific sizes of the superclusters spines is different
from that found for the other standard statistics. Furthermore, it is also
shown that at higher redshifts ($z\ge 0.3$) the deviations of the mean specific
sizes of the supercluster spines for all of the four considered combinations
from its value for the standard $\Lambda$CDM case are statistically
significant.
