general relativity - Can a curvature singularity (i.e. BH), as defined in terms of geodesic incompleteness, actually exist in nature?

A curvature invariant is a scalar representation of curvature derived from a curvature tensor. The classic example is the Kretschmann scalar derived from the Riemann curvature, where $K=R_{μνλρ}R^{μνλρ}$.

This is a coordinate independent measure that allows one to discern between coordinate and curvature singularities. For example in the Schwarzschild black hole spacetime the Kretschmann scalar is

$$K=R_{μνλρ}R^{μνλρ}=\dfrac{48M^2}{r^6}$$

where $M$ is the geometrized mass.

Different geometries yield different functions for the curvature invariants and since all classical BH geometries are required to have singularities by the Geodesic Incompleteness theorems of Hawking and Penrose, then the curvature necessarily runs to infinity. For the example above it should be obvious that

$$\lim_{r\to0}K=\lim_{r\to0}\dfrac{48M^2}{r^6}=\infty$$

There are no anti-infinity theorems. If so, can a curvature singularity, as in geodesic incompleteness, actually exist in nature?