Evenness conjecture for the Hurwitz continued-fraction invariant density

Let $h$ be the invariant density, viewed as a function of $x+y\mathbbm{i}$ with real coordinates $x,y$. Consider the points $\pm.5\pm.5\mathbbm{i}$, $\pm.5$, and $\pm.5\mathbbm{i}$. Evenness conjecture. The Taylor series of $h$ is even in both the $x$ and $y$ coordinates around every one of these points. Equivalently, if $x_0+y_0\mathbbm{i}$ is any such point, then

$$\left.\frac{\partial^{m+n}}{\partial^m x\partial^n y}h(x+y\mathbbm{i})\right|_{x+y\mathbbm{i}=x_0+y_0\mathbbm{i}}=0$$

whenever $m$ or $n$ is odd. This conjecture is suggested by numerical calculations of the invariant density and its apparent rotational symmetry, but the statement remains unproved in the supplied text.