Bounded acyclicity of Diff(D^n, ∂) in higher regularity

Determine whether the group Diff(D^n, ∂) of orientation-preserving C^r diffeomorphisms of the n-disc fixing the boundary pointwise is boundedly acyclic for all regularities r ≥ 1 and dimensions n ≥ 3; that is, show that H^k_b(Diff(D^n, ∂)) = 0 for every k > 0 in this range of (r, n).

Background

The paper proves bounded acyclicity for Homeo(D^n, ∂) in all dimensions, extending previous results from dimension 2 and relying on a controlled annulus theorem. However, the authors’ methods currently address only the topological (C^0) category in higher dimensions, and they do not establish the analogous result in the smooth (C^r, r ≥ 1) category.

This question asks to extend bounded acyclicity—vanishing of bounded cohomology with real coefficients in all positive degrees—to smooth diffeomorphism groups of discs with fixed boundary for higher regularities and dimensions, where differentiability at the boundary introduces additional technical challenges.