This question already has an answer here:
Consider a star, for simplicity a non rotating one. The corresponding spacetime will be similar to a Schwartzschild one (if the star is static and spherically symmetric).
Outside the star we will have exactly the Schwartzschild fall-off, while inside the star we will not have a horizon or a singularity, since the matter content of a sphere contained in the star is proportional to the sphere volume itself.
If this star collapses into a black hole, for the same reason I would not expect the black hole to necessarily contain a naked singularity.
I would say that the distribution of matter related to the black hole is all inside the Schwartzschild radius, but couldn't it be that once you enter it the matter spatial distribution prevent the singularity at the center?
The argument seems very similar to the one used in the stellar case.
I think that any motivation on the line of: "inside the horizon of a Schwartzschild metric all the geodesics will crush into the singularity" should be reviewed because you could have a non point-like matter distribution inside and a non Schwartzschild metric, like in the stellar case.
I think that maybe, if you have even the thinnest shell of empty space right inside the horizon photons there will start falling and therefore anything else at smaller radii will be falling, hence bringing to a pointlike singularity.
But even if that was correct, I still think we could have cases where there is no empty space inside the horizon.
So, I would very much appreciate any insight on how are we sure that star collapse in General Relativity brings to a singularity. If it is true, how can it be fully proved? And in particular, where the reasoning in the case of a BH with no empty space would fail?