The boundary-support conjecture for the local time of super-Brownian motion

Let $S(X_t)=\overline{\{x:X(t,x)>0\}}$ be the closed support of $X_t$, let $\partial S(X_t)$ denote its boundary, and let $L_t$ be the local-time measure associated with the boundary of the zero set. Here $\lambda_0$ is the constant appearing in the dimension estimate for the boundary of the zero set. Boundary-support conjecture. The measure $L_t$ is supported on $\partial S(X_t)$ and, consequently,

$$\dim(\partial S(X_t))=2-2\lambda_0$$

on $\{X_t\neq 0\}$, $P_{X_0}^X$-almost surely and $\mathbb{N}_0$-almost everywhere. This would identify the support of the local time with the boundary of the support of the super-Brownian motion and determine the boundary's Hausdorff dimension on the non-extinction event. The claim is presented as a conjecture because it is not established whether $BZ_t\setminus\partial S(X_t)$ is non-empty; in particular, the existence of isolated zeros remains unresolved.