Mottram's path-selection conjecture for conditioned Brownian motion

Let $U\subseteq\mathbbm{R}\times[0,\infty)$ be open with $(0,0)\in U$, and let $\operatorname{cost}(P)$ be the path cost. Assume that there is a unique path $P$ from $(0,0)$) to $\partial U$ such that $\operatorname{cost}(P)<\operatorname{cost}(\widetilde P)$ for every other such path $\widetilde P$. Let $\mathcal{E}_a^U$ be the corresponding conditioning events. Mottram's path-selection conjecture. The path $P$ is parametrised by $(vt,t)$ for a constant $v$, the conditioned measures $\mathbbm{W}(\mathord\cdot\mid\mathcal{E}_a^U)$ converge weakly to a limit $\mathbbm{Q}^U$ as $a\to\infty$, and

$$\frac{W_t}{t}\longrightarrow v$$

in $\mathbbm{Q}^U$-probability. The conjecture formalises the expectation that a unique least-cost exit path determines the macroscopic route and limiting speed; the supplied context gives supporting cost estimates but no proof of the full assertion.