The main conjecture on rigid isotopy classes of quartic Morse polynomials

Let $f\colon\mathbb{R}^2\to\mathbb{R}$ be a Morse polynomial of degree $4$, and consider its principal part. The principal part may be non-discriminant positive definite, vanish on two real lines, or vanish on four real lines. Main conjecture. There are exactly $13$, respectively $29$ and $16$, rigid isotopy classes of such Morse polynomials in the three cases, as listed in the relevant table and realized in the remainder of the article. The conjecture gives the expected complete count of rigid isotopy classes for the three types of principal part; the preceding theorem establishes the available classification results and the remaining assertion is supported by the constructions in the article.