Sullivan [S] proved that on any $n$-dimensional, $n\neq 4$, topological manifold there is a unique Lipschitz structure. Although it is generally accepted result, it seems that the proof lacks some details making it difficult to understand, see: Well known theorems that have not been proved.

On the other hand Donaldson and Sullivan [DS] proved that there are smooth $4$-dimensional manifolds that are not bi-Lipschitz homeomoerphic, so the Lipschitz structure in $4$-manifolds, even if it exists, is not unique.

Question. Is the Lipschitz structure on $\mathbb{S}^4$ unique?

I guess the answer must be well known. I think I know how to prove it using some results of Sullivan, but since I am not an expert in this field, my argument could be wrong and I would not even be able to see where I am making a mistake.

[DS] S. K. Donaldson, D. P. Sullivan,, Quasiconformal 4-manifolds. Acta Math. 163 (1989), 181-252.

[S] D. Sullivan, Hyperbolic geometry and homeomorphisms. Geometric topology (Proc. Georgia Topology Conf., Athens, Ga., 1977), pp. 543–555, Academic Press, New York-London, 1979.