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 ∼1.4M_⊙. In this work, we investigate the relation between the radii of very massive neutron stars up to the maximum mass, M_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_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_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.