On martingale convergence

Let $(X_t)_{t\ge0}$ be a martingale with continuous paths. It was previously shown here and here that then it is impossible that $X_t\to\infty$ almost surely as $t\to\infty$.

Is it possible that there exist a nonrandom sequence $(t_k)$ in $[0,\infty)$ converging to $\infty$ such that $X_{t_k}\to\infty$ almost surely as $k\to\infty$? That is,

do there exist a martingale $(X_t)_{t\ge0}$ with continuous paths and a nonrandom sequence $(t_k)$ in $[0,\infty)$ converging to $\infty$ such that $X_{t_k}\to\infty$ almost surely as $k\to\infty$?