Existence and identification of the limiting local-time measure

For a fixed interval $[a,b]$, let $\mathsf{\overline{\mu}}_r[a,b]$ be the finite-radius approximations to the local-time measure and let $\mathsf{\overline{\mu}}[a,b]$ denote their prospective limit. Local-time convergence conjecture. The $L^2$-limit

$$\overline{\mu}[a,b]=\lim_{r\to\infty}\overline{\mu}_r[a,b]$$

exists, and then, by the basic convergence theorem, $\mathsf{\overline{\mu}}=\mu$ almost surely. Establishing this convergence identifies the approximating local-time measures with the limiting local time of dynamical percolation. The source provides no resolution of the conjecture.