I am certainly no physicist but I remember coming across academic papers in the past speculating about exactly your question. I recall one theorized about singularities being hollow with all of the mass (err was it space? spacetime?) compacted down into 2 dimensions on a shell at the surface (at least IIUC, which I probably didn't).

I think that concept might fit with the infinite time dilation preventing a merger from ever actually occurring? I'd be curious how that might differ for matter that's already inside when the critical mass is reached. (I'd also be curious to know all the creative and wacky ways in which I got the above completely wrong given that's just about inevitable.)