Subcritical heat kernels decay faster than critical heat kernels

Let $M$ be a connected noncompact Riemannian manifold, and let $k_P^M(x,y,t)$ denote the positive minimal heat kernel of a second-order elliptic operator $P$ on $M$. An operator is subcritical if its minimal Green function is finite off the diagonal, and critical otherwise. Let $P_+$ and $P_0$ be respectively subcritical and critical operators in $M$. Subcritical–critical heat-kernel ratio conjecture.

$$\lim_{t\to\infty}\frac{k_{P_+}^M(x,y,t)}{k_{P_0}^M(x,y,t)}=0$$

locally uniformly in $M\times M$. The claim is known for positive-critical $P_0$ and for certain short-range Schrödinger operators, but the general case remains open.