Stevens's conjecture on simple curve singularities

Let a reduced complex curve singularity be called simple when all singularities occurring in its versal deformation fall into finitely many equivalence classes. A parametrisation of a curve singularity is simple when it has the analogous finiteness property for neighbouring parametrisations.

Stevens's conjecture. The simple reduced curve singularities are exactly those with simple parametrisation.

In embedding dimension three, the classifications of simple parametrisations and simple curves coincide, while beyond embedding dimension three the classification of simple curves is largely unknown. The conjecture would in particular resolve the question of whether rigid reduced curve singularities exist, since any such singularity would be simple.