Extremal-distance distribution conjecture for first-passage local sets

Let $\Sigma$ be a domain with boundary pieces $B_1(\Sigma)$, $B_2(\Sigma)$ and $B_3(\Sigma)$, and let $\Lambda_a^\Sigma$ be the minimal local set containing $B_1(\Sigma)$ whose conditional Gaussian free field has boundary value $a$ on the local set, value $h_2$ on $B_2(\Sigma)\setminus\Lambda_a^\Sigma$, and free boundary condition on $B_3(\Sigma)\setminus\Lambda_a^\Sigma$. Assume $a<\min_{B_1(\Sigma)}h_1$, and let $m$ be the excursion-measure average of the boundary value $h_1$ on $B_1(\Sigma)$.

First-passage extremal-distance conjecture. The extremal distance $\operatorname{ED}(\Lambda_a^\Sigma,B_2(\Sigma))$ is distributed as the last visit time of level $a$ by a Brownian bridge of length $\operatorname{ED}(B_1(\Sigma),B_2(\Sigma))$ from $h_2$ to $m$, with the last visit time defined to be $0$ if the bridge never hits $a$.

This extends the metric-graph first-passage identity to continuum Gaussian free fields and local sets. The statement is presented as a conjectural scaling-limit consequence of the corresponding metric-graph result; its general continuum validity remains open.