The stable homeomorphism theorem says that:
Every orientation preserving and surjective homeomorphism $f:\mathbb{R}^n\to\mathbb{R}^n$ can be written as $f=f_1\circ\ldots\circ f_k$, where $f_i:\mathbb{R}^n\to\mathbb{R}^n$ are surjective homeorphisms and $f_i|_{U_i}=\operatorname{id}$ for some open set $\varnothing\neq U_i\subset\mathbb{R}^n$.
It is not difficult to see that it follows that the same result holds for homeomorphisms of $\mathbb{S}^n$, because any surjective homeomorphism of $\mathbb{R}^n$ can be identified with a homeomorphism of $\mathbb{S}^n$ that fixes the north pole. For a nice and elementary introduction to stable homeomorphisms see [BG] (written when the stable homeomorphism theorem was still a conjecture).
Sullivan [S] "proved" the stable homeomorphism theorem for bi-Lipschitz homeomorphisms.
Tukia and Väisälä [TV] tried to fill missing details in Sullivan's theorey, but they were not able to verify one important step and this is why I put "proved" in inverted commas, see Well known theorems that have not been proved. In fact they proved the following result (assuming existence of Sullivan manifolds):
Every orientation preserving and surjective locally bi-Lipschitz homeomorphism $f:\mathbb{R}^n\to\mathbb{R}^n$ can be written as $f=f_1\circ\ldots\circ f_k$, where $f_i:\mathbb{R}^n\to\mathbb{R}^n$ are surjective locally bi-Lipschitz homeorphisms and $f_i|_{U_i}=\operatorname{id}$ for some open set $U_i\subset\mathbb{R}^n$.
My question is:
Question. Is it true that if $f:\mathbb{S}^n\to\mathbb{S}^n$ is orientation preserving and bi-Lipschitz, then it can be written as $f=f_1\circ\ldots\circ f_k$, where $f_i:\mathbb{S}^n\to\mathbb{S}^n$ are bi-Lipschitz homeorphisms and $f_i|_{U_i}=\operatorname{id}$ for some open set $\varnothing\neq U_i\subset\mathbb{S}^n$?
I am asking this question, because it would allow one to extend most of the results in [BG] to the bi-Lipschitz category and in particular (as far as I checked the details) it would imply that the stable bi-Lipschitz homeomorphism theorem implies the bi-Lipschitz annulus theorem with a completely elementary argument.
[BG] Brown, M., Gluck, H.: Stable structures on manifolds. I: Homeomorphis of $\mathbb{S}^n$. Ann. Math. 79, (1964) 1-17.
[S] Sullivan, D.: Hyperbolic geometry and homeomorphisms. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 543–555, Academic Press, New York-London, 1979.
[TV] Tukia, P.; Väisälä, J.: Lipschitz and quasiconformal approximation and extension. Ann. Acad. Sci. Fenn. Ser. A I Math. 6 (1981), no. 2, 303–342 (1982).
