I'm interested in the behaviour of the stopped $\alpha$-Hölder norm of a one-dimensional real-valued Brownian motion $(B_t)_{t \geq 0}$ for $\alpha < 1/2$.

For fixed $T>0$, self similarity implies that $$\Bbb E\left[ \sup_{0 \leq s < t \leq T} \frac{|B_t-B_s|}{|t-s|^\alpha}\right] = T^{1/2 - \alpha}\Bbb E\left[ \sup_{0 \leq s < t \leq 1} \frac{|B_t-B_s|}{|t-s|^\alpha}\right].$$

Does the corresponding upper bound extend to the stopped Hölder norm? In other words, does there exist a random integrable constant $C$ such that for all bounded stopping times $\tau > 0$ (w.r.t. the sigma algebra generated by $B_t$), $$\Bbb E\left[ \sup_{0 \leq s < t \leq \tau} \frac{|B_t-B_s|}{|t-s|^\alpha}\right] \leq \Bbb E [C \tau^{1/2 - \alpha}],$$ where $C$ depends on $\alpha < 1/2$?

Thanks in advance.