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Very good. The problem is that singularities are quantum objects. Quantum physics works nothing like classical physics.
For example, in the case of perpetually falling singularities, would they just quantum tunnel into each other? Or would singularities even exist? According to general relativity, singularities are a sphere that never stops being compressed due to its own gravity. What happens when this sphere hits a diameter smaller than Plank's length? Does the universe take a screenshot? The point is, we have absolutely no clue about what's happening here.
To understand the above, we would first need to understand how gravity works at the quantum level, which we don't. Why? Gravity is incredibly weak. Studying it is thus, very hard.