Prescribing Ricci curvature on a Product of Spheres

We prove an existence result for the prescribed Ricci curvature equation for certain doubly warped product metrics on $mathbb{S}^{d_1+1}times mathbb{S}^{d_2}$, where $d_i geq 2$. If $T$ is a metric satisfying certain curvature assumptions, we show that $T$ can be scaled independently on the two factors so as to itself be the Ricci tensor of some metric.