A super-Brownian representation conjecture for non-removable sets

Let $\Sigma\subset \mathbb{R}^d$ be a compact set that is not a removable set for $\Delta$ in $L^2(\mathbb{R}^d)$. A self-adjoint extension of $\Delta|_{C_c^\infty(\mathbb{R}^d\backslash\Sigma)}$ in $L^2(\mathbb{R}^d)$ is an operator $\widetilde{\Delta}$, and let $\{\mathcal{X}_t\}_{t\geq 0}$ be a measure-valued process. Super-Brownian representation conjecture. There exist such an extension $\widetilde{\Delta}$ and process $\{\mathcal{X}_t\}_{t\geq 0}$ such that

$$\mathtt{E}_{\delta_x}[\mathcal{X}_t(u_0)]=\widetilde{u}(t,x)$$

for any $t>0$ and any $x\in\mathbb{R}^3\backslash\Sigma$, where $\widetilde{u}$ is the solution of

$$\partial_tu=\frac{1}{2}\widetilde{\Delta}u,\qquad u(0,x)=u_0(x).$$

This is presented as a plausible generalization of a result of Fleischman and Mueller, extending the connection between removability, super-Brownian motion, and capacities to non-removable sets. The source does not provide evidence that the conjecture has been resolved.