There are a number of caveats I think I can see at this point. In the first place Black Holes -- or, I suppose more accurately, black hole-like objects -- have been observed, so that there are things existing with black hole properties is not really in dispute. In the second place the paper considers only a specific and idealised model, so as per usual might not capture the real world precisely. This last is always a problem in Science, of course -- one just hopes that the approximation is "close enough". In this case a more accurate model may not display the explosive behaviour.
And in the third place all sorts of interesting stuff can go on mathematically that turns out to have no relevance to the physical world. What this paper shows, if it shows anything, is that the threshold between classical General Relativity and QM is even more bizarre than had previously been appreciated. It's possible -- indeed, almost certain -- that a more complete theory that fuses these two successfully would manage to find a way around this problem that allows black holes to form without approaching the singularity. In this sense, it's possible that this result is closer in nature to such papers as, say, the Rayleigh-Jeans paper of 1905.
I suppose I should briefly explain that one, although the story can be found in most popular histories of QM -- the result basically shows that in classical thermodynamics, the radiation of "black bodies" should blow up at high frequencies. Clearly it did not. The conclusion to be drawn from the paper, then, was that the then current understanding of physics was wrong. This was resolved five years earlier by Planck's work that laid the foundations for Quantum Theories (this timing is often forgotten -- Rayleigh's and Jeans' work proved the failure of old physics and was vital in allowing the new to be accepted).
If this paper is as ground-breaking, it will prove the failure of forced GR/ QM hybrid theories and provide a further reason to find a better understanding of Quantum Gravity.