Classification conjecture for compact reducible Riemannian manifolds with conformal product structures

Let $(M,g)$ be a compact connected Riemannian manifold with reducible holonomy, and let $D$ be a conformal product structure on $(M,g)$ different from the Levi-Civita connection of $g$. Classification conjecture. The manifold $(M,g)$ is either conformally flat or a triple product. Here, a triple product is a Riemannian manifold locally isometric to a product construction of three manifolds, with metric $e^{2f}g_1+g_2+g_3$ on $M_1\times M_2\times M_3$, where $f$ is a smooth function on $M_1\times M_2$. The conjecture is presented as evidence toward a classification of compact reducible Riemannian manifolds carrying non-Levi-Civita conformal product structures; the supplied text does not establish whether the proposed dichotomy is proved or remains open.