The homogeneity conjecture for supergravity backgrounds

Let $(M,g,F)$ be a supergravity background, and let $\nu$ denote its fraction of preserved supersymmetry. The background is homogeneous if it admits a transitive action of a Lie group via $F$-preserving isometries.

Homogeneity conjecture. All supergravity backgrounds with

$$\nu>\tfrac12$$

are homogeneous.

The conjecture proposes a critical supersymmetry fraction above which a supergravity background must be homogeneous. Maximally supersymmetric backgrounds are symmetric, whereas the M2-brane has $\nu=\tfrac12$ and cohomogeneity one; known solutions with larger supersymmetry fraction were reported to be homogeneous, but the general assertion remains open.