The conjecture on rigid singularities

Let $(X,x)$ be a singularity. A singularity is rigid if every deformation is trivial. A fat point is a zero-dimensional nonreduced singularity; a reduced curve singularity is a singularity on a reduced curve; and a normal surface singularity is a normal two-dimensional singularity.

Rigidity conjecture. There exist no rigid fat points, no rigid reduced curve singularities, and no rigid normal surface singularities.

The existence of rigid singularities in small dimension is stated as an open problem; the conjecture concerns the absence of such singularities in the listed classes.