Arnold's structural stability conjecture for finite-parameter families on the two-sphere

Let a finite-parameter family of vector fields on $S^2$ be given, and call it weakly topologically equivalent to another family when a germ of a homeomorphism of parameter spaces makes corresponding vector fields orbitally topologically equivalent.

Arnold's conjecture. Generic finite-parameter families of vector fields on the two-sphere are structurally stable: close families are weakly topologically equivalent.

Arnold's conjecture was disproved for moderate equivalence; the supplied source specifically states that the disproof in that setting does not address weak equivalence, so the status of the weak-equivalence formulation is not established here.