Smooth-branch conjecture for generic curve singularities

Let $C$ be a generic curve singularity, in the sense proposed in the surrounding discussion: it is parametrised by a generic point of a component of a base space of a versal deformation, with the relevant genericity conditions stated there. A branch is an irreducible local component of $C$. Smooth-branch conjecture. Every branch of $C$ is smooth. The paper presents this as a conjecture based on its examples, and the excerpt gives no proof or resolution.