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 $Sigma _{i}m_{ u _{i}}$ 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 $Sigma _{i}m_{ u _{i}}$ we find that -- at the present epoch -- the value of $Sigma _{i}m_{ u _{i}}$ 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 ($zge 0.3$) 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.
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