Mottram's local-time exponent conjecture for ballistic limiting measures

Let $v>\gamma^{\circ}$ and let $\mathbbm{Q}^{U_v}=\lim_{a\to\infty}\mathbbm{W}(\mathord\cdot\mid\mathcal{E}_a^{U_v})$ be any measure supplied by the critical-speed framework. Let $L_x(\infty)=\lim_{T\to\infty}L_x(T)$. Mottram's ballistic local-time exponent conjecture. There exists a constant $C_v>0$ such that, as $\varepsilon\to0$,

$$\lim_{x\to\infty}\mathbbm{Q}^{v}\bigl(L_x(\infty)>1-\varepsilon\bigr)\sim C_v\varepsilon^3.$$

The exponent $3$ follows heuristically from the proposed stationary local-time distribution and is proved for the model distribution used in the heuristic, but the assertion for every ballistic limiting measure remains open.