Cosmic Censorship:
Weak or strong, the cosmic censorship conjecture states that naked singularities can’t be seen, otherwise everything will break down, it would be really bad and worst of all theorists would be confused.
Hawking and Ellis, in The LargeScale Structure of Space-Time (Cambridge 1973)
But it turns out that singularities very likely don’t actually exist in a real universe governed by GR. Any lumpy, non symmetric space time can have all the spinning black holes it wants – at any angular momentum, even with a > m (angular momentum greater than the mass in suitable units), as the Kerr solution + bumps (bumps are incoming GR full bandwidth noise), will have no paths leading to any singularity! So the curtain can be lifted, the horizon is not needed to protect us.
Cosmic Serendipity Conjecture:
In any sufficiently complex solution of GR, there exists no singularities. I am not talking about naked singularities here, I mean any and all singularities.
The complex nature of the interaction of GR 
at the tiny scales where the singularity would start to form stop that very formation. In other words, the singularity fails to form as the infalling energy always has some angular momentum in a random direction, and ruins the formation of a singularity.
In all likelihood actual physical spinning black holes in a turbulent environment (normal space) will have no singularity.
I will let Brandon Carter speak now:
“Thus we reach the conclusion that at timeline or null geodesic or orbit cannot reach the singularity under any circumstances except in the case where it is confined to the equator, cos() = 0…..Thus as symmetry is progressively reduced, starting from the Schwarchild solution, the extent of the class of geodesics reaching the singularity is steadily reduced likewise, … which suggests that after further reduction in symmetry, incomplete geodesics may cease to exist altogether”
Kerr Fields, Brandon Carter 1968.
Not cosmic censorship, but almost the opposite – singularities can’t exist in an GR universe (one with bumps) because there are no paths to them.
We have all been taught that singularities form quickly – that when a non – spherical mass is collapsing, GR quickly smooths the collapse, generating a singularity, neatly behind a horizon. Of course that notion is correct, but what it fails to take into account is that in a real situation, there is always more in falling energy, and that new infalling energy messes up the formation of the singularity.
While there may be solutions to Einstein’s equations that show a singularity (naked or not), these solutions are unphysical, in that the real universe is bumpy and lumpy. So while the equations hold ‘far’ away from the singularity, the detailed Gravity in the high curvature region keeps it just that – high curvature as opposed to a singularity.
The papers of A.Burinskii come to mind, e.g.:
Kerr Geometry as Space-Time Structure of the Dirac Electron
Conclusion
I am willing to bet that this conjecture is experimentally sound, in that there are no experiments that have been done to refute it. (that’s a joke I think).
On the theory side, one would have to prove that a singularity is stable against perturbation by incoming energy, which from my viewpoint seems unlikely, as the forming singularity would have diverging fields and diverging response to incoming energy, which would blow it apart. Like waves in the ocean that converge on a rocky point.
–Tom
Andersen, T. C. on ORCID
General Relativity 