Neumann–Pichon's conjecture on metrically conical complex analytic sets

Let $X\subset\mathbb{C}^n$ be a complex analytic set with isolated singularity at $0$. A set is metrically conical at $0$ if, for some $\varepsilon>0$, its germ at $0$ is bi-Lipschitz equivalent to the cone over its link by a norm-preserving homeomorphism; it is locally metrically conical at $0$ if the corresponding conical property holds near every direction in the link of its tangent cone.

Neumann–Pichon's conjecture. $X$ is metrically conical at $0$ if and only if $X$ is locally metrically conical at $0$.

The paper presents this as a conjecture about the relationship between global metric conicalness of an isolated complex analytic singularity and the corresponding local property. The stated theorem gives a partial answer by characterizing smoothness in terms of local metric conicalness together with high connectivity of the link.