Reducible Riemannian manifolds with conformal product structures

We study conformal product structures on compact reducible Riemannian manifolds, and show that under a suitable technical assumption, the underlying Riemannian mani\-folds are either conformally flat, or triple products, \emph{i.e.} locally isometric to Riemannian manifolds of the form $(M,g)$ with $M=M_1\times M_2\times M_3$ and $g=e^{2f}g_1+g_2+g_3$, where $g_i$ is a Riemannian metric on $M_i$, for $i\in\{1,2,3\}$, and $f\in C^\infty(M_1\times M_2)$.