Finite diffeomorphism types in dimension four without the Kähler condition under Ricci lower bound and L2-Ricci bound

Establish whether there exists a constant C(D,Λ,V) > 0 such that the class M(D,Λ,V) := {(M^4,g) closed: diam(M) ≤ D, Vol(M) ≥ V, Ric ≥ −3, and ∫_M |Ric|^2 ≤ Λ} contains at most C(D,Λ,V) many diffeomorphism types.

Background

After proving finiteness of diffeomorphism types for compact 4-dimensional Kähler manifolds with Ricci ≥ −3, volume noncollapsing, bounded diameter, and L2 scalar curvature bound (Theorem 1.2), the authors pose the natural extension to the non-Kähler setting. They define the class M(D,Λ,V) of closed 4-manifolds with Ricci ≥ −3, Vol ≥ V, diam ≤ D, and ∫|Ric|^2 ≤ Λ.

The question asks whether a universal bound C(D,Λ,V) exists on the number of diffeomorphism types within this class. The authors note that, based on their methods, it would suffice to show that tangent cones of limits of sequences in M(D,Λ,V) are cones over smooth cross sections with a uniform lower bound on the Reifenberg radius, which would yield bounded L2-curvature via Chern–Gauss–Bonnet.