Component-count conjecture for $J_{10}^3$ singularities

Let $f_0$ be a real function singularity of type $J_{10}^3$. Consider the discriminant variety in the parameter space of any versal deformation of $f_0$.

$J_{10}^3$ component conjecture. The complement of the discriminant variety has exactly 33 distinct components.

The preceding proposition describes 23 virtual components, their involution, and ten additional components obtained by reflection in a vertical mirror. The claim is that these constructions account for all actual components of the discriminant complement; the source also notes that topological-type invariants distinguish all but two pairs.