Chavel's conjecture on Neumann heat-kernel monotonicity

Let $D_1,D_2\subset \mathbb{R}^d$ be smooth bounded convex domains, with $d\geq 1$. For $i=1,2$, let $p_{D_i}(t,x,y)$ denote the Neumann heat kernel in $D_i$. If $D_2\subset D_1$, then Chavel's conjecture.

$$p_{D_1}(t,x,y)\leq p_{D_2}(t,x,y)$$

for every $t\geq 0$ and $x,y\in D_1$. The conjecture predicts that, unlike the Dirichlet heat kernel, the Neumann heat kernel is monotone in the reverse direction under domain inclusion; the supplied source does not establish its resolution.