Let $B=(B_t)_{0\le t\le T}$ be a continuous semi-martingale and $\mathbb F=(\mathcal F_t)_{0\le t\le T}$ be its natural filtration. Denote by $\mathcal C_b(\Omega\times \mathbb R_+)$ the space of continuous bounded functions $F:\Omega\times \mathbb R_+ \to \mathbb R$, where $\Omega$ denotes the space of continuous functions defined on $[0,T]$ endowed with the uniform norm, and $\Omega\times\mathbb R_+$ is equipped with the product topology.

Now, consider a sequence of $\mathbb F$-stopping times $(\tau_n)_{n\ge1}$ s.t. there exists a random variable $\tau$ satisfying

$$\lim_{n\to \infty}\mathbb E[F(B,\tau_n)]~~=~~\mathbb E[F(B,\tau)],~~ \mbox{ for all } F\in\mathcal C_b(\Omega \times \mathbb R_+).$$

My question is: could we show that $\tau$ is an $\mathbb F$-stopping time? If not, what about assuming that $B$ is Markov or even a Brownian motion? Thanks a lot for the reply!