Mottram's critical-speed conjecture for Brownian entropic repulsion

Let $\gamma^{\circ}$ and $\gamma^{\bullet}$ be the critical speeds with $1<\gamma^{\circ}<\gamma^{\bullet}$, and let $\mathcal{E}_a^{U_v}$ be the conditioning events associated with an open set $U_v$. Mottram's critical-speed conjecture. There exists $\gamma^{\circ}$ such that for every $v>\gamma^{\circ}$ there is an open set $U_v$ for which

$$\mathbbm{Q}^{U_v}=\lim_{a\to\infty}\mathbbm{W}(\mathord\cdot\mid\mathcal{E}_a^{U_v})$$

exists and satisfies

$$\lim_{t\to\infty}\frac{W_t}{t}=v$$

in $\mathbbm{Q}^{U_v}$-probability, whereas for every $v<\gamma^{\circ}$ no such open set exists. The threshold is motivated by comparing the cost of direct travel with the cost of a path that travels faster and returns; the existence and nonexistence assertions remain conjectural.