Critical-subcritical heat-kernel comparison conjecture

Let $M$ be a Riemannian manifold, and let $P_+$ and $P_0$ be respectively subcritical and critical operators in $M$. Critical-subcritical decay conjecture. The ratio

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

locally uniformly in $M\times M$. The conjecture expresses the expectation that the heat kernel of a subcritical operator decays faster than that of a critical operator. The source presents it in general settings; no resolution is supplied here.