The homogeneous harmonic-curvature conjecture

A homogeneous Riemannian manifold is a Riemannian manifold $M$ whose isometry group acts transitively. It has harmonic curvature when its curvature tensor $R$ has vanishing divergence, equivalently when its Ricci tensor is Codazzi:

$$\nabla_X(\operatorname{Ric})(Y,Z)=\nabla_Y(\operatorname{Ric})(X,Z)$$

for all vector fields $X,Y,Z$. It is Ricci-parallel when $\nabla(\operatorname{Ric})=0$.

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

The conjecture is motivated by the absence, to the authors' knowledge, of non-Ricci-parallel homogeneous Riemannian manifolds with harmonic curvature. The paper proves the assertion for left-invariant metrics on solvable Lie groups and for Lie groups of dimension at most $6$, while the general homogeneous case remains open in the supplied source.