  Prospects of establishing the radii of massive neutron stars in PSR
J1614-2230 and PSR J0740+6620 from NICER and Chandra observatories hold the
potential to constrain the equation of state (EoS) of matter to densities well
beyond those encountered in canonical stars of mass $\sim 1.4\,{\rm
M}_{\odot}$. In this work, we investigate the relation between the radii of
very massive neutron stars up to the maximum mass, $M_{\rm max}$, supported by
dense matter EoSs. Results from models with hadronic matter are contrasted with
those that include a first-order hadron-to-quark phase transition. We find that
a lower bound on $M_{\rm max}$ with an upper bound on the radius of massive
pulsars serves to rule out too soft quark matter, and an upper bound on $M_{\rm
max}$ with a lower bound on the radius of massive pulsars strongly disfavors a
transition into too-stiff quark matter appearing at low densities. The
complementary role played by radius inferences from future gravitational wave
events of inspiraling binary neutron stars is also briefly discussed.
