  Following the idea that a stable sexaquark state with quark content (uuddss)
would have gone unnoticed by experiment so far and that such a particle would
be a good dark matter candidate, we investigate the possible role of a stable
sexaquark in the physics of compact stars given the stringent constraints on
the equation of state that stem from observations of high mass pulsars and
GW170817 bounds on the compactness of intermediate mass stars. We find that
there is a "sexaquark dilemma" (analogous to the hyperon dilemma) for which the
dissociation of the sexaquark in quark matter is a viable solution fulfilling
all present constraints from multi-messenger astronomy. The parameters needed
to model the hybrid star including sexaquarks are in line with parameters of
pre-existing quark- and hadronic-matter models. We find that current
constraints -- tidal deformability in accordance with GW170817 and maximum mass
above the lower limit from PSR J0740+6620 -- can be satisfied two ways: with
early quark deconfinement such that neither sexaquarks nor hyperons are present
in any NS interiors, or with later deconfinement such that a neutron-sexaquark
shell surrounds the inner quark matter core.
