The splitting conjecture for non-compact Einstein manifolds with cocompact isometry groups

Let $(M^n,g)$ be a complete negative Einstein manifold admitting a proper, isometric, cocompact action by a connected Lie group.

Splitting conjecture. $M^n$ splits isometrically as a product of a compact Einstein manifold and a non-compact homogeneous Einstein space.

This proposes that the unimodularity assumption in the preceding splitting theorem is unnecessary. The statement extends the known result for connected unimodular acting groups, while the general case remains open.