Certainly not a black hole! That would not be stable situation at all.
The content of the core of a neutron star is the subject of much speculation. The possibilities fall into a number of categories. (i) An increasingly hard neutron equation of state, such that neutrons retain their identities as they are squeezed closer, but an increasingly repulsive many-body strong nuclear force provides support. (ii) Additional hadronic degrees of freedom, such that the neutrons (and protons) transform into other heavy hadrons such as lambda or sigma particles. (iii) Some sort of quark plasma. (iv) Boson condensations involving the neutrons decaying into pions or kaons with zero momentum.
There are a number of diagnostics of these possibilities: primarily, the maximum possible mass of a neutron star should decrease from about 3 solar masses for (i) down to about 1.5 solar masses for (iv). Secure measurements of a 2 solar mass neutron star would seem to rule out (iv), but even that seems not completely agreed. Another diagnostic is how quickly neutron stars can cool. The presence of quark matter or boson condensations should lead to much more rapid cooling by neutrino emission. Again, nothing conclusive has emerged yet.
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