The all-dimensional $L^2$ curvature bound conjecture

Let $M^n$ be a noncollapsed Riemannian manifold with bounded Ricci curvature, namely $|\operatorname{Ric}_{M^n}|\leq n-1$ and $\operatorname{Vol}(B_1(p))>{\rm v}>0$.

All-dimensional $L^2$ curvature bound conjecture. There exists $C=C(n,{\rm v})>0$ such that

$$\fint_{B_1(p)}|{\rm Rm}|^2\leq C.$$

The paper proves the corresponding estimate in dimension $4$, improving the bounds for all $q<2$ to an a priori $L^2$ bound. Whether the same estimate holds in every dimension is left as a conjecture.