Lê's conjecture on injective holomorphic map germs

Let $f\colon (\mathbb{C}^n,0)\to (\mathbb{C}^{n+1},0)$ be an injective holomorphic map germ. A map germ is corank $1$ when the differential $df_0$ has corank $1$. Lê's conjecture. The map germ $f$ is corank $1$. This conjecture concerns the differential rank of injective holomorphic map germs and is posed after results relating injectivity, smoothness, holomorphic embedding, and Lipschitz normal embedding. Its status is not determined by the supplied source context.