Ambient Lipschitz equivalence conjecture for normally embedded real surface germs

Let $(S_0,0)$ and $(S_1,0)$ be two normally embedded real semialgebraic surface germs. They are normally embedded when their inner and outer metrics are bi-Lipschitz equivalent; for such sets, the two metrics are equivalent. Ambient Lipschitz equivalence conjecture. If $S_0$ and $S_1$ are ambient topologically equivalent and bi-Lipschitz equivalent with respect to either the inner or outer metric, then they are ambient Lipschitz equivalent. The conjecture asks whether metric equivalence together with ambient topological equivalence determines ambient Lipschitz equivalence for normally embedded real semialgebraic surface germs. The surrounding discussion exhibits non-normally embedded examples where the conclusion fails, so the normally embedded hypothesis is essential; no resolution is supplied here.