Ragsdale's conjecture for $M$-curves

Let $C$ be an $M$-curve of even degree $m=2k$, with $p$ even ovals and $n$ odd ovals, and let $\chi=p-n$ be the Euler characteristic of the Ragsdale membrane. Ragsdale's $M$-curve conjecture. One has

$$|\chi|\le k^2.$$

The general Ragsdale conjecture was refuted, but the restriction to $M$-curves is stated as still open in the source.