Bounded acyclicity of Diff(S^n)

Determine whether the group Diff(S^n) of orientation-preserving C^r diffeomorphisms of the n-sphere is boundedly acyclic for all regularities r ≥ 0 and dimensions n ≥ 2; explicitly, show that H^k_b(Diff(S^n)) = 0 for every k > 0 in this range of (r, n).

Background

The authors establish bounded acyclicity for various transformation groups, including Homeo(R^n) and Homeo(D^n, ∂), and derive isomorphisms relating discs and spheres in bounded cohomology in certain cases. Nevertheless, beyond low-degree computations, the bounded cohomology of Homeo(S^n) and Diff(S^n) is largely unknown for n ≥ 2.

This question asks for a complete vanishing result in bounded cohomology for the smooth diffeomorphism group of spheres across regularities and dimensions, extending understanding well beyond the presently known dimension 1 case.