Neumann heat kernel monotonicity along hyperbolic radii in symmetric convex domains

Let $\Omega$ be a bounded convex domain in the plane symmetric with respect to the $x$-axis, and let $p^N_{\Omega}(t,z,w)$ be its Neumann heat kernel. Let $f:\mathbb{B}\to\Omega$ be a conformal map from the unit disk onto $\Omega$ such that the image of the diameter joining $-1$ and $1$ is the axis of symmetry of $\Omega$. Hyperbolic-radius monotonicity conjecture. For every $t>0$, the diagonal $p^N_{\Omega}(t,f(z),f(z))$ is strictly increasing along each hyperbolic radius intersecting the horizontal axis: if $z_1=r_1e^{i\theta}$ and $z_2=r_2e^{i\theta}$, with $0<\theta<\pi$ and $0<r_1<r_2\leq 1$, then

$$p^N_{\Omega}(t,f(z_1),f(z_1))<p^N_{\Omega}(t,f(z_2),f(z_2)).$$

This generalizes the ball monotonicity question to symmetric convex planar domains and is motivated by its connection with the hot-spots problem. No resolution is supplied in the source.