Birbrair–Gabrielov conjecture on bi-Lipschitz equivalence of LNE surface germs

Let $(X_0,0)$ and $(X_1,0)$ be two LNE semialgebraic surface germs. They are ambient topologically equivalent if there is an ambient homeomorphism taking one germ to the other, and outer bi-Lipschitz equivalent if such an ambient map can be chosen bi-Lipschitz for the outer metric.

Birbrair–Gabrielov conjecture. If $(X_0,0)$ and $(X_1,0)$ are ambient topologically equivalent and outer bi-Lipschitz equivalent, then $X_0$ and $X_1$ are ambient bi-Lipschitz equivalent.

The conjecture is false: the paper's construction of LNE Hölder triangles in $\mathbb R^4$ with microknots gives a negative answer, producing examples that satisfy the stated topological and outer bi-Lipschitz equivalences but are not ambient bi-Lipschitz equivalent.