Uniform scalar-curvature conjecture for four-dimensional shrinking Ricci solitons

Let $\bar{E}>0$, and let $(M,g,f)$ be a four-dimensional shrinking Ricci soliton whose Euler characteristic satisfies $|\chi(M)|\leq \bar{E}$. Write $\mathcal{R}_g$ for the scalar curvature of $g$. Uniform scalar-curvature conjecture. There exists $\bar{S}>0$, depending only on $\bar{E}$, such that

$$\sup_M \mathcal{R}_g\leq \bar{S}.$$

A bound of this kind would provide a uniform geometric estimate for four-dimensional shrinking Ricci solitons with bounded Euler characteristic. The paper presents it as a conjectural expectation motivated by analogous results for self-shrinkers, and gives no resolution.