Here are some results for running the BE problem past sink formation:

From the BP collapse (quarter of the grid):

On the left side is a plot of a zoomed in section of the BE sphere (sink formation is at frame = 96). You can see the density compression wave converging on the center. In the right panel are the sinks. At first one forms, then a next, and a next, to a multitude. This is not what we'd expect. What we should expect is exact symmetry: at the center of the collapsing sphere a sink should form, stay stationary, and accrete the surrounding envelope. My assumption is that this is not happening because of an even number of cells forcing the sink to form off center of the potential (due to AMR and the quadrant of the BE sphere), exacerbating asymmetry in the mesh and subsequent evolution.

Except I realized this wasn't a great way to see what was happening… Here is a better view —

From the Matched collapse:

Here is early (pre sink) zoomed out evolution:

Here is late zoomed in (sink and post sink formation):