Bounded cohomology of diffeomorphism groups of higher dimensional spheres

In this paper we prove the vanishing of the bounded cohomology of $\text{Diff}^r_+(S^n)$ with real coefficients when $n\geq 4$ and $1\leq r\leq \infty$. This answers the question raised in \cite{FNS24} for $\geq 4$ dimensional spheres.