Vanishing odd-derivative conjecture for the $V_{3,1}$ Hurwitz density

Let $h_{m,n}^{3,1}(x,y)$ denote the coefficients or derivatives used for the invariant density associated with the region $V_{3,1}$, evaluated at the center $(x,y)$. Vanishing odd-derivative conjecture.

$$h_{m,n}^{3,1}(-.5,0)=0$$

whenever $m$ or $n$ is odd. The natural symmetry of $V_{3,1}$ explains evenness in the $y$ direction, while the additional vanishing of odd $x$-derivatives is suggested by the numerical calculations; the supplied text does not establish it.