Minimality conjecture for the ground state of a critical parabolic operator

Let $P$ be a critical operator in a Riemannian manifold $\mathcal{M}$, and let $\varphi$ denote its ground state. Let $\mathcal{H}_P(\mathcal{M}\times\mathbb{R})$ be the cone of positive solutions associated with the parabolic operator $P$ on $\mathcal{M}\times\mathbb{R}$. Ground-state minimality conjecture. The ground state $\varphi$ is a minimal positive solution in the cone

$$\mathcal{H}_P(\mathcal{M}\times\mathbb{R}).$$

The conjecture was posed by the author in the cited earlier work; the supplied text gives no resolution, so its general status remains open.