Even-lacuna conjecture for isolated plane critical points

Let $f(x_1,x_2)$ have an isolated non-Morse critical point at $0$, and let its deformations be small deformations of this function. An even local lacuna is a local connected component of the complement of the discriminant in parameter space on which the even Petrovskii class vanishes. A critical value means the value of a perturbed function at one of its real critical points.

Even-lacuna conjecture. The deformations of $f$ have no even local lacunas unless $f$ has an extremum at the origin. If $f$ has a minimum at the origin, then all critical values of real critical points of all small perturbations belonging to such a lacuna are positive; if $f$ has a maximum, they are negative.

This claim would characterize the possible even local lacunas for isolated non-Morse critical points in two variables. The supplied context gives the extremum example but does not state whether the conjecture has been proved or disproved.