Parity condition for simple Hurwitz numbers

Let $g\ge 0$ and let $\mu$ be a partition with $|\mu|\ge 3$. Write $h_{g,\mu}$ for the corresponding simple Hurwitz number, $r(g,\mu)$ for its associated ramification number, and $\ell(\mu)$ for the length of $\mu$. Parity condition. If $h_{g,\mu}$ is odd, then $r(g,\mu)$ is even, every part of $\mu$ is odd, and

$$\ell(\mu)\le 2.$$

This condition records a parity restriction found in the study of simple Hurwitz numbers. The supplied text does not indicate whether the condition has been proved or remains open.