The morsification–mutation conjecture for plane curve singularities

Let $(C,z)$ and $(C',z')$ be real isolated plane curve singularities, and choose real morsifications of them. Each morsification has an associated quiver; two quivers are mutation equivalent when one can be transformed into the other by iterated quiver mutations. The morsification–mutation conjecture. The singularities $(C,z)$ and $(C',z')$ have the same complex topological type if and only if their associated quivers are mutation equivalent. This conjecture proposes a classification of complex topological types using the quivers of real morsifications; the paper presents it as a hypothetical answer to the problem of recognizing when different real forms and morsifications represent the same complex singularity.