A genuine problem I think about sometimes is reconciling the 5d Gregory Laflamme process with the classical no-splitting theorem for black holes (I believe this was brought up in this context by Giddings). The situation is that a line-extended 5d black hole is unstable to splitting into spherical holes, but these can't properly separate in the classical theory, they are linked by tendrils of horizon cylinders that get narrow and presumably connect the black holes to each other classically indefinitely. Gubser analyzed this situation a decade or so ago, looking for a stable end-point, but it seems obvious that there is no stable endpoint, that the black holes must separate completely on physical grounds, theorem or no theorem.
To understand what is going wrong with the classical theorem (it is correctly proved), there is an an analogous paradox in continuum mechanics for viscous fluids (I see it in my electronic cigarette every day!). If you take a viscous fluid, and try to separate a blob into two separate drops, you just can't do it in continuum-mechanics, and for real viscous fluids (like e-juice) you end up having a thin tendril filament linking the two drops forever, and it just becomes more and more tenuous. But the atomic structure of the matter leads the tendril to eventually statistically break, separating the drops.
I strongly suspect that similarly, the ultimately quantum properties of the black hole is breaking the black holes of the end stage of the Gregory Laflamme instability into separate holes. There might be a semi-stable tendril for a long time, as in the viscous fluid drop-separation case, or the tendrils might classically want to become infinitesimal right from the start. It requires study to know.
There are few possibilities for exact solutions to help here (although it is possible to study something similar using exact solutions). But numerical methods are perfect for this problem, as there is cylindrical symmetry and translational symmetry which reduces the computational load to managable levels. You can put it in a box, and do expansions for the solution. So as a numerical GR question, I think it is ideal. A problem regarding this is that the thin tendrils require a nonuniform method of describing the solution, like a nonuniform lattice, as is usual in numerical GR, where the regions of high field are fast-varying and the regions of low field are slow varying. There are wavelet methods that might work here, I haven't studied it in detail, it's just something on the to-do list. It is very accessible problem, and it probably requires quantum black holes to fully understand the process, much as the analogous fluid problem requires atomism for the tearing process of the fluid tendril.