@j Somehow this smells a bit like BS, because this is the first time I've heard anything about black holes requiring anything that could be described as "exotic matter".

Also, the entire singularity problem has kinda been debunked before. You see, if you do the math with Einstein's equations, and assume a static (non-rotating) black hole, you get a singularity as a result of the equations. But to date, all observed black holes have been found to be spinning.
Doing the math on a spinning black hole is much, much harder, but apparently you don't get a singularity anymore from Einstein's equations. You still get an event horizon, a point beyond which the gravity is so large that no light can escape, but underneath that the matter is shaped into a highly dense donut shape. No physics breaking, infinitely small, singularity required.

Of course, Einstein's theory is considered incomplete, and people are looking and hoping for a quantum gravity theory to know for sure what happens in these extreme conditions.