Besse's spherical CPE conjecture

Let $(M^n,g,f)$ be a CPE metric, meaning that $(M^n,g)$ is a closed, oriented Riemannian manifold of dimension $n\geq 3$ with constant scalar curvature and $f:M^n\to\mathbb{R}$ is a nonconstant smooth potential function satisfying

$$\nabla^2 f=(1+f)\widetilde{Ric}_g-\frac{R}{n(n-1)}fg.$$

Here $\widetilde{Ric}_g$ is the traceless Ricci tensor. Besse's spherical CPE conjecture. A CPE metric with a nonconstant solution $f$ is isometric to a round sphere metric. This is the refined spherical form of the CPE conjecture, motivated by the fact that the round sphere was the only known nonconstant example in the source; it remains open in general.