Let $(M,g)$ be a closed, smooth, Riemannian manifold of dimension $n>4$. Suppose both the scalar curvature and norm of the Ricci tensor are constant. In addition suppose that $g$ satisfies the following condition: $(\frac{1}{2} \Delta Ric_{ij} + Ric^{ml}W_{milj} - \frac{3n}{4(n-1)^2}Ric_{ij} -\frac{2}{n-2}Ric^l_j Ric_{il}) = \lambda g_{ij}.$

Here $r$ is the scalar curvature, $Ric$ is the Ricci tensor, $W$ is the Weyl tensor, $\Delta = g^{-1}\nabla\nabla$, and $\lambda$ is a real number. This condition is the Euler-Lagrange equation of a quadratic Riemannian functional. Notice that Einstein metrics satisfy it. Can one conclude that $g$ is Einstein?