The CPE conjecture on critical metrics being Einstein

A CPE metric is a 3-tuple $(M^n,g,f)$, where $(M^n,g)$ is a compact oriented Riemannian manifold of dimension $n\geq 3$ with constant scalar curvature, and $f$ is a smooth function satisfying

$$\operatorname{Ric}-\frac{R}{n}g=\operatorname{Hess} f-\left(\operatorname{Ric}-\frac{R}{n-1}g\right)f.$$

Here $\operatorname{Ric}$, $R$, and $\operatorname{Hess}$ denote the Ricci tensor, scalar curvature, and Hessian, respectively. CPE conjecture. Every CPE metric is Einstein. This conjecture is the CPE formulation of the conjecture that the critical points of the total scalar curvature functional restricted to the space of constant-scalar-curvature metrics are Einstein. The source gives no resolution, so the conjecture remains open.