  We reconsider the problem of the hyperon puzzle and its suggested solution by
quark deconfinement within the two-phase approach to hybrid compact stars with
recently obtained hadronic and quark matter equations of state. For the
hadronic phase we employ the hypernuclear equation of state from the lowest
order constrained variational method and the quark matter phase is described by
a sufficiently stiff equation of state based on a color superconducting
nonlocal Nambu-Jona-Lasinio model with constant (model nlNJLA) and with
density-dependent (model nlNJLB) parameters. We study the model dependence of
the phase transition obtained by a Maxwell construction. Our study confirms
that also with the present set of equations of state quark deconfinement
presents a viable solution of the hyperon puzzle even for the new constraint on
the lower limit of the maximum mass from PSR J0740+6620. In this work we
provide with model nlNJLB for the first time a hybrid star EoS with an
intermediate hypernuclear matter phase between the nuclear and color
superconducting quark matter phases, for which the maximum mass of the compact
star reaches $2.2~M_\odot$, in accordance with most recent constraints. In
model nlNJLA such a phase cannot be realised because the phase transition onset
is at low densities, before the hyperon threshold density is passed. We discuss
possible consequences of the hybrid equation of state for the deconfinement
phase transition in symmetric matter as it will be probed in future heavy-ion
collisions at FAIR, NICA and corresponding energy scan programs at the CERN and
RHIC facilities.
