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We infer the collapse times of long-lived neutron stars into black holes using the x-ray afterglows of 18 short gamma-ray bursts. We then apply hierarchical inference to infer properties of the neutron star equation of state and dominant spin-down mechanism. We measure the maximum non-rotating neutron star mass MTOV=2.31-0.21+0.36M⊙ and constrain the fraction of remnants spinning down predominantly through gravitational-wave emission to η=0.69-0.39+0.21 with 68% uncertainties. In principle, this method can determine the difference between hadronic and quark equation of states. In practice, however, the data is not yet informative with indications that these neutron stars do not have hadronic equation of states at the 1σ level. These inferences all depend on the underlying progenitor mass distribution for short gamma-ray bursts produced by binary neutron star mergers. The recently announced gravitational-wave detection of GW190425 suggests this underlying distribution is different from the locally measured population of double neutron stars. We show that MTOV and η constraints depend on the fraction of binary mergers that form through a distribution consistent with the locally measured population and a distribution that can explain GW190425. The more binaries that form from the latter distribution, the larger MTOV needs to be to satisfy the x-ray observations. Our measurements above are marginalized over this unknown fraction. If instead, we assume GW190425 is not a binary neutron star merger, i.e., the underlying mass distribution of double neutron stars is the same as observed locally, we measure MTOV=2.26-0.17+0.31M⊙. [ABSTRACT FROM AUTHOR]