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 $n\ge 4$ and $1\le r\le \mathrm{\infty }$. This answers the question raised in \cite{FNS24} for $\ge 4$ dimensional spheres.