Full-support conjecture for the dynamical percolation local time

Let $\mathsf{\mathscr{E}}$ be the exceptional time set and let $\mathsf{\mu}$ be the associated local-time measure, which is supported inside $\mathsf{\mathscr{E}}$. Full-support conjecture. The support of the local time measure $\mathsf{\mu}$ is almost surely the entire exceptional time set $\mathsf{\mathscr{E}}$. This would show that every exceptional time is detected by the local-time measure, not merely that the measure is concentrated on the exceptional set. The source provides no resolution of this conjecture.