Existence conjecture for real morsifications of plane curve singularities

Let $(C,z)$ be a real plane curve singularity. A real morsification is a real deformation of $(C,z)$ whose nearby singularities are ordinary double points and whose real part has the required morsification properties.

Existence conjecture. Every real plane curve singularity possesses a real morsification.

The preceding theorem establishes this for singularities satisfying an admissibility condition along each non-real tangent line at every real point of the minimal resolution tree. The conjecture asks whether the same existence statement holds without that restriction.