Cao's rigidity conjecture for gradient shrinking Ricci solitons

Let $(M,g,f)$ be a gradient shrinking Ricci soliton with constant scalar curvature. In the terminology recalled in the source, such a soliton is rigid when it is a flat bundle $N\times_{\Gamma}\mathbb R^k$, where $N$ is an Einstein manifold, $\Gamma$ acts freely on $N$ and by orthogonal transformations on $\mathbb R^k$, and $f=\frac{\lambda}{2}|x|^2$ on $\mathbb R^k$. Cao's conjecture. Every gradient shrinking Ricci soliton with constant scalar curvature is rigid.

Peterson and Wylie proved that a gradient Ricci soliton is rigid if and only if it has constant scalar curvature and is radially flat, namely $\operatorname{Rm}(\nabla f,\cdot,\nabla f,\cdot)=0$. Cao's conjecture asks whether the radial-flatness condition follows automatically from constant scalar curvature in the shrinking case; its resolution is not given in the provided text.