Strong ratio-limit conjecture for subcritical and critical heat kernels

Let $M$ be a Riemannian manifold. Let $P_+$ and $P_0$ be respectively subcritical and critical elliptic operators in $M$, and let $k_{P_+}^M(x,y,t)$ and $k_{P_0}^M(x,y,t)$ denote their positive minimal heat kernels.

Strong ratio-limit conjecture. Locally uniformly for $(x,y)\in M\times M$,

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

The conjecture predicts a strong long-time separation between the heat kernels of subcritical and critical operators. The source presents it as a far-reaching open conjecture attributed to M. Fraas, D. Krejčiřík and Y. P.