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The nonlocal three-flavor Nambu-Jona-Lasinio model is used to study quark deconfinement in the cores of neutron stars (NSs). The quark-hadron phase transition is modeled using both the Maxwell construction and the Gibbs construction. For the Maxwell construction, we find that all NSs with core densities beyond the phase transition density are unstable. Therefore, no quark matter cores would exist inside such NSs. The situation is drastically different if the phase transition is treated as a Gibbs transition, resulting in stable NSs whose stellar cores are a mixture of hadronic matter and deconfined quarks. The largest fractions of quarks achieved in the quark-hadron mixed phase are around 50%. No choice of parametrization or composition leads to a pure quark matter core. The inclusion of repulsive vector interactions among the quarks is crucial since the equation of state (EoS) in the quark-hadron mixed phase is significantly softer than that of the pure hadronic phase.