The conjecture that homogeneous Riemannian manifolds with harmonic curvature are Ricci-parallel

Let $M$ be a homogeneous Riemannian manifold. It has harmonic curvature when the divergence of its curvature tensor vanishes; equivalently, its Ricci tensor $\operatorname{Ric}$ is Codazzi.

Homogeneous harmonic-curvature conjecture. Any homogeneous Riemannian manifold $M$ with harmonic curvature is Ricci-parallel:

$$\nabla(\operatorname{Ric})=0.$$

Einstein and Ricci-parallel metrics provide immediate examples of manifolds with harmonic curvature. The conjecture is known for nilpotent and solvable Lie groups in the cited results, as well as for Lie groups of dimension at most $6$, but remains open in general.