Conformal product structures on compact Einstein manifolds

In this note we generalize our previous result, stating that if $(M_1,g_1)$ and $(M_2,g_2)$ are compact Riemannian manifolds, then any Einstein metric on the product $M:=M_1\times M_2$ of the form $g=e^{2f_1}g_1+e^{2f_2}g_2$, with $f_1\in C^\infty(M_2)$ and $f_2\in C^\infty(M_1\times M_2)$, is a warped product metric. Namely, we show that the same conclusion holds if we replace the assumption that the manifold $M$ is globally the product of two compact manifolds by the weaker assumption that $M$ is compact and carries a conformal product structure.