Conjecture on variance lower bounds for white-noise Schrödinger operators

Let $D709$ be a white noise, and suppose that Assumption D709 holds. In Cases 1 and 2, let $V$ be a potential on $I$ such that, for some $D709,D708>0$,

$$V(x)\leq D709|x|+D708\qquad\text{for all }x\in I.$$

White-noise semigroup variance conjecture. One has

$$\liminf_{t\to0}\operatorname{Var}\bigl[\operatorname{Tr}[\hat K(t)]\bigr]>0.$$

This conjecture concerns the optimal conditions under which the variance of the heat-semigroup trace vanishes as $t\to0$ for one-dimensional continuous random Schrödinger operators. It is proposed specifically for white noise in Cases 1 and 2, but the supplied context does not define Assumption Potential, the cases, or the operator $\hat K(t)$ explicitly.