  Can one distinguish a binary black hole undergoing a merger from a binary
neutron star if the individual compact companions have masses that fall inside
the so-called mass gap of $3-5\ M_\odot$? For neutron stars, achieving such
masses typically requires extreme compactness and in this work we present
initial data and evolutions of binary neutron stars initially in
quasiequilibrium circular orbits having a compactness $C=0.336$. These are the
most compact, nonvacuum, quasiequilibrium binary objects that have been
constructed and evolved to date, including boson stars. The compactness
achieved is only slightly smaller than the maximum possible imposed by
causality, $C_{\rm max}=0.355$, which requires the sound speed to be less than
the speed of light. By comparing the emitted gravitational waveforms from the
late inspiral to merger and postmerger phases between such a binary neutron
star vs a binary black hole of the same total mass we identify concrete
measurements that serve to distinguish them. With that level of compactness,
the binary neutron stars exhibit no tidal disruption up until merger, whereupon
a prompt collapse is initiated even before a common core forms. Within the
accuracy of our simulations the black hole remnants from both binaries exhibit
ringdown radiation that is not distinguishable from a perturbed Kerr spacetime.
However, their inspiral leads to phase differences of the order of $\sim 5$ rad
over an $\sim 81$ km separation (1.7 orbits) while typical neutron stars
exhibit phase differences of $\geq 20$ rad. Although a difference of $\sim 5$
rad can be measured by current gravitational wave laser interferometers (e.g.,
aLIGO/Virgo), uncertainties in the individual masses and spins will likely
prevent distinguishing such compact, massive neutron stars from black holes.
