  We investigate the hadron-quark phase transition in cold neutron stars in
light of (i) the observed limits on the maximum-mass of heavy pulsars, (ii)
constraints on the tidal properties inferred from the gravitational waves
emitted in binary neutron-star mergers, and (iii) mass and radius constraints
derived from the observation of hot spots on neutron star observed with NICER.
Special attention is directed to the possible presence of $\Delta(1232)$
baryons in neutron star matter. Our results indicate that this particle could
make up a large fraction of the baryons in neutron stars and thus have a
significant effect on the properties of such objects, particularly on their
radii. This is partially caused by the low density appearance of $\Delta$s for
a wide range of theoretically defensible sets of meson-hyperon, SU(3) ESC08
model, and meson-$\Delta$ coupling constants. The transition of hadronic matter
to quark matter, treated in the 2SC+s condensation phase, is found to occur
only in neutron stars very close to the mass peak. Nevertheless, quark matter
may still constitute an appreciable fraction of the stars' total matter if the
phase transition is treated as Maxwell-like (sharp), in which case the neutron
stars located beyond the gravitational mass peak would remain stable against
gravitational collapse. In this case, the instability against gravitational
collapse is shifted to a new (terminal) mass different from the maximum-mass of
the stellar sequence, giving rise to stable compact objects with the same
gravitational masses as those of the neutron stars on the traditional branch,
but whose radii are smaller by up to 1 km. All models for the equation of state
of our study fall comfortably within the bound established very recently by
Annala {\it et al.} (Nature Physics, 2020)
