Krejčiřík and Zuazua's heat-semigroup decay conjecture

Let $\Omega$ be an open connected subset of $\mathbb{R}^d$. Let $H_0$ and $H_+$ be two self-adjoint operators in $L^2(\Omega)$ such that

$$\inf \sigma(H_0)=\inf \sigma(H_+)=0.$$

Assume that there is a positive function $\rho:\Omega\to\mathbb{R}$ such that $H_+\geq\rho$, while $H_0-V$ is a negative operator for any non-negative non-trivial $V\in C_0^\infty(\Omega)$. Krejčířík and Zuazua's conjecture. There exists a positive function $K:\Omega\to\mathbb{R}$ such that

$$\lim_{t\to\infty}\frac{\lVert e^{-tH_+}\rVert_{L^2(\Omega,K)\to L^2(\Omega)}}{\lVert e^{-tH_0}\rVert_{L^2(\Omega,K)\to L^2(\Omega)}}=0.$$

The conjecture predicts that a positive lower-order term produces asymptotically faster heat-semigroup decay under the stated criticality assumptions. The source paper proves it for locally sheared unbounded strips.