Equivalence conjecture for heat kernels under compact perturbations

Let $M$ be a Riemannian manifold, and let $P_1$ and $P_2$ be subcritical elliptic operators of the form specified in the paper. Two heat kernels are equivalent when they are comparable above and below by positive constants on $M\times M\times(0,\infty)$. Suppose that $P_1=P_2$ outside a compact subset of $M$. Compact-perturbation heat-kernel equivalence conjecture. The heat kernels $k_{P_1}^M$ and $k_{P_2}^M$ are equivalent. Equivalence of heat kernels under compact perturbations is known only in limited cases, so this conjecture remains open.