  High-density cores (HDCs) of galaxy superclusters that embed rich clusters
and groups of galaxies are the earliest large objects to form in the cosmic
web, and the largest objects that may collapse in the present or future. We
study the dynamical state and possible evolution of the HDCs in the BOSS Great
Wall (BGW) superclusters at redshift $z \approx 0.5$ in order to understand the
growth and evolution of structures in the Universe. We derived the density
contrast values for the spherical collapse model in a wide range of redshifts
and used these values to study the dynamical state and possible evolution of
the HDCs of the BGW superclusters. The masses of the HDCs were calculated using
stellar masses of galaxies in them. We found the masses and radii of the
turnaround and future collapse regions in the HDCs and compared them with those
of local superclusters. We determined eight HDCs in the BGW superclusters. The
masses of their turnaround regions are in the range of $M_{\mathrm{T}} \approx
0.4 - 3.3\times~10^{15}h^{-1}M_\odot,$ and radii are in the range of
$R_{\mathrm{T}} \approx 3.5 - 7 h^{-1}$Mpc. The radii of their future collapse
regions are in the range of $R_{\mathrm{FC}} \approx 4 - 8h^{-1}$Mpc. Distances
between individual cores in superclusters are much larger: of the order of $25
- 35h^{-1}$Mpc. The richness and sizes of the HDCs are comparable with those of
the HDCs of the richest superclusters in the local Universe. The BGW
superclusters will probably evolve to several poorer superclusters with masses
similar to those of the local superclusters. This may weaken the tension with
the $\Lambda$CDM model, which does not predict a large number of very rich and
large superclusters in our local cosmic neighbourhood, and explains why there
are no superclusters as elongated as those in the BGW in the local Universe.
