Let $B_t$ be a standard Brownian motion and $L_t^x$ its local time in $x$ at time $t$. For fixed $t$ and $p>1$, it holds that $$ \sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] < \infty,$$ which can for example be proven by $$\sup_{x \in \mathbb{R}} \operatorname{E} [ (L_t^x)^p ] \leq \operatorname{E} [ \sup_{x \in \mathbb{R}} (L_t^x)^p] \leq C_p \, \operatorname{E} [ \sup_{0 \leq s \leq t} |B_t|^p ] \leq C_p \operatorname{E} [ \langle B,B \rangle_t^{p/2} ] = C_p t^{p/2},$$ where $C_p$ is a positive constant which changes between inequalities. A proof for the second inequality (which also holds for $t=\infty$) can be found in "(Semi-)Martingale Inequalities and Local Times" by Barlow/Yor from the eighties, the third inequality is Burkholder-Davis-Gundy.
My first question: Am I correct in assuming that one can replace $t$ by a $L^1$-bounded stopping time and the supremum remains finite? In the proof above, the first inequality stays valid, the second and third should follow from stopping but I don't know how to bound $\operatorname{E} [\langle B_{\tau},B_{\tau} \rangle^{p/2}]$.
My second question: Does it still hold that the supremum is finite if $\tau$ is just assumed to be almost surely finite?