Hernquist distribution of matter as a source of black-hole geometry

It was recently demonstrated that imposing the condition $P_{r} = -\rho$ on the radial pressure of a galactic halo can lead to regular black-hole solutions for certain density profiles, such as the Dehnen and Einasto models. In the present work, we show that some of the most commonly used halo profiles, including the Hernquist model, do not yield regular geometries under the same condition, but instead support black-hole solutions that retain a central singularity