The supremum of Brownian local times on Hölder curves, II

For $f:[0,1]\to \mathbb{R}$, we consider $L_t^f$, the local time of space-time Brownian motion on the curve $f$. Let ${\mathcal{S}}_{\alpha }$ be the class of all functions whose Hölder norm of order $\alpha$ is less than or equal to 1. We show that the supremum of $L_1^f$ over $f$ in ${\mathcal{S}}_{\alpha }$ is finite if $\alpha >\frac{1}{2}$.