Ground-state-normalized Dirichlet heat kernel monotonicity in the ball

Let $\mathbb{B}$ be the unit ball in $\mathbb{R}^d$, with $d\geq 1$, let $p^D_{\mathbb{B}}(t,x,y)$ be the Dirichlet heat kernel, and let $\varphi_1(x)$ be the ground-state eigenfunction for the Laplacian in $\mathbb{B}$ with Dirichlet boundary conditions. Ground-state-normalized monotonicity conjecture. For every $t>0$, the radial function

$$\frac{p^D_{\mathbb{B}}(t,x,x)}{\varphi_1^2(x)}$$

is strictly increasing with $|x|$: whenever $0\leq |x_1|<|x_2|\leq 1$,

$$\frac{p^D_{\mathbb{B}}(t,x_1,x_1)}{\varphi_1^2(x_1)}<\frac{p^D_{\mathbb{B}}(t,x_2,x_2)}{\varphi_1^2(x_2)}.$$

The claim is motivated by Dirichlet heat-kernel monotonicity and by conditioned Brownian-motion analogues of hot-spots results. The source gives no resolution.