Laugesen–Morpurgo conjecture for the Neumann heat kernel of the unit ball

Let $p_{\mathbb{U}}(t,x,y)$ denote the heat kernel for the Laplacian with Neumann boundary conditions on the unit ball

\mathbb{U}=\left\\{x\in\mathbb{R}^{n}:\left\lVert x\right\rVert<1\right\\}

in $\mathbb{R}^{n}$, where $n\in\mathbb{N}^{\ast}$. The function $p_{\mathbb{U}}(t,x,x)$ is the diagonal element of the heat kernel. Laugesen–Morpurgo conjecture. For every $t>0$, $p_{\mathbb{U}}(t,x,x)$ is strictly increasing as a function of $\lVert x\rVert$: if $x,y\in\mathbb{U}$ satisfy $\lVert x\rVert<\lVert y\rVert$, then

$$p_{\mathbb{U}}(t,x,x)<p_{\mathbb{U}}(t,y,y).$$

The paper states that probabilistic arguments settle this conjecture for all dimensions, so the claim is solved rather than open. The result concerns radial monotonicity of the Neumann heat kernel and is connected with the Hot Spots problem for Neumann eigenfunctions.