Let $\ell^n: [0,\infty)\to [0,1]$ be right-continuous and increasing functions s.t. $\ell^n(0)=0$. Given $x>0$ and Brownian motion $(B_t)_{t\ge 0}$, can we prove $$\limsup_{n\to\infty}\mathbb P[\exists s\in [0,t]:~ x+B_s\le \ell^n(s)]\le \mathbb P[\exists s\in [0,t]:~ x+B_s\le \limsup_{n\to\infty}\ell^n(s)],\quad \forall t>0?$$

PS : It appears that the pathwise inequality $$\limsup_{n\to\infty} {\bf 1}_{\{\exists s\in [0,t]: x+B_s\le \ell^n(s)\}}\le {\bf 1}_{\{\exists s\in [0,t]: x+B_s\le \limsup_{n\to\infty}\ell^n(s)\}}$$ does not hold.