A positive lower bound for the scalar curvature of non-flat Ricci shrinkers

Let $n\geq 4$ be a positive integer, and let $(M,p,g,f)$ be a Ricci shrinker. Denote its scalar curvature by $R$, and write $\sup_{x\in M}R(x)$ for its supremum. Uniform scalar-curvature lower-bound conjecture. There exists a constant $\epsilon=\epsilon(n)>0$ such that, for every non-flat Ricci shrinker $(M,p,g,f)$,

$$\sup_{x\in M}R(x)\geq\epsilon.$$

This conjecture asserts a dimension-dependent gap away from zero for the maximum scalar curvature of non-flat Ricci shrinkers. The supplied text gives no resolution.