The Decomposition Problem for bi-Lipschitz maps of spheres

Let $LIP(\mathbb{S}^n)$ denote the orientation-preserving homeomorphisms of the $n$-sphere that are bi-Lipschitz. For a bi-Lipschitz map, its isometric distortion is the least $L\geq 1$ such that the map is $L$-bi-Lipschitz. Decomposition Problem. Let $n\geq 1$ and let $f\in LIP(\mathbb{S}^n)$. Then for every $\epsilon>0$ there are an integer $m\geq 1$ and homeomorphisms $f_k\in LIP(\mathbb{S}^n)$, for $k=1,\ldots,m$, such that

$$f=f_m\circ\cdots\circ f_1,$$

with each $f_k$ having isometric distortion at most $1+\epsilon$. The problem asks whether every bi-Lipschitz map of a sphere admits such a finite decomposition into maps with arbitrarily small distortion; the supplied context does not establish whether the assertion is known in full generality.