Davies' strong ratio limit conjecture for heat kernels

Let $Lu=u_t+P(x,\partial_x)u$ be a parabolic operator defined on a noncompact Riemannian manifold $M$, with $\lambda_0(P,M)\geq 0$. Fix a reference point $x_0\in M$. Davies' conjecture. The limit

$$\lim_{t\to\infty}\frac{k_P^M(x,y,t)}{k_P^M(x_0,x_0,t)}=a(x,y)$$

exists and is positive for all $x,y\in M$. Davies proposed this in the selfadjoint case; it concerns whether the large-time heat-kernel decay is approached equally fast at different points. The conjecture was disproved in the discrete setting by Kozma, while the stated manifold version is not resolved here.