In this section, we briefly remark on some possible extensions of the results of this paper. To begin with, we recall that one of the main applications of this paper was to combine the codimension $4$ estimates of Theorem 1.1 with the ideas of quantitative stratification in order to show for all $q<2$ that $\fint_{B_{1}(p)}|{\rm Rm}|^{q}$ is uniformly bounded when $M^{n}$ is a noncollapsed manifold with bounded Ricci curvature. Furthermore, in dimension $4$ we were able to improve this to show a bound on $\fint_{B_{1}(p)}|{\rm Rm}|^{2}$. We conjecture that this holds in any dimension.

In a different direction, another main result of the paper was to show that in dimension $4$, noncollapsed manifolds with bounded diameter and Ricci curvature have finite diffeomorphism type. In higher dimensions, this is too much to hope for; see for instance [HN14] where noncollapsed Calabi-Yau manifolds of real dimension $\geq 6$ are constructed with unbounded third Betti number. Nonetheless, we conjecture that under the assumption of bounded Ricci curvature, one should expect a bound on the second Betti number.

Note that examples of Menguy and Perelman show that it is actually necessary to assume a $2$-sided bound on the Ricci tensor; see [Men2000].