The correspondence conjecture for strictly Morse quartic polynomials

Let $f:\mathbb R^2\to\mathbb R$ be a Morse polynomial of degree $4$ with nine real critical points, and fix a rigid isotopy class of such polynomials. A polynomial is strictly Morse when distinct critical points have distinct critical values. The D-graph of the rigid isotopy class has a standard partial order on its vertices.

Correspondence conjecture. For any rigid isotopy class, the isotopy classes of strictly Morse polynomials contained in it are in a natural one-to-one correspondence with the extensions of the standard partial order of the vertices of its D-graph to a total order.

Strictly Morse classification refines rigid classification by separating equal critical values. If the conjecture holds, the number of strictly Morse classes is bounded above by the corresponding $\operatorname{Card}$ invariant divided by $10$, the number of possible numbers of negative critical values. The supplied text gives no resolution status.