Let $W^1, W^2, \ldots$ be i.i.d. Brownian motions, $T_1, T_2, \ldots$ be random variables converging a.s. to $t > 0$, do we have CLT for $W^n_{T_n}$?

Question

Let $W^1, W^2, \ldots$ be i.i.d. Brownian motions, $T_1, T_2, \ldots$ be non-negative random variables converging a.s. to $t > 0$. Does CLT $$ \frac{1}{\sqrt {nt}} (W^1_{T_1} + \cdots + W^n_{T_n}) \xrightarrow[n\to\infty]{\mathcal D} \mathcal N_{0,1} $$ hold?

In this case $\left\{W^n_{T_n}\right\}$ is not an independent sequence, but, I speculate, asymptotically i.i.d. in some sense. Do we have existing results for this type of sequence?