Korchagin's parity conjecture for degree-9 M-curves with three nests

Let $C_9$ be an $M$-curve of degree $9$ with real scheme

$$\langle \mathcal{J} \amalg 1 \langle \alpha_1 \rangle \amalg 1 \langle \alpha_2 \rangle \amalg 1 \langle \alpha_3 \rangle \amalg \beta\rangle.$$

Here, the integers $\alpha_i$ count the empty ovals in the three nests. Korchagin's conjecture. At least two of the $\alpha_i$, for $i=1,2,3$, are odd. This conjecture concerns the still-open classification of isotopy types of real plane $M$-curves of degree $9$; the paper proves only that at least one of the three numbers $\alpha_i$ is odd, excluding 41 isotopy types.