My poor internal physicist thinks this is unlikely. When you read about density of black holes, it is often mentioned that the supermassive ones aren't very den

My poor internal physicist thinks this is unlikely. When you read about density of black holes, it is often mentioned that the supermassive ones aren't very dense (in the sense of mass/Schwarzschild volume). I take it the smaller a black hole is, the higher its density must be. Therefore it should be unlikely for much smaller parts of a star that does not form a black hole as a whole to become black holes as they must achieve much higher density than that of the star.