Uniform integrability of process with bounded conditional expectation

Question

Let $[0, T]$ be a finite time horizon, i.e., $T < \infty$. Consider a complete filtered probability space $(\Omega, {\cal F}, {\mathbb F}, P)$, where ${\mathbb F} = \{ {\cal F}_t \}_{t \in [0, T]}$ is a Brownian filtration and satisfies the usual conditions. Let $\{X_t\}_{t \in [0, T]}$ be an ${\mathbb F}$-adapted process with continuous sample path. If ${\mbox E} [ X_\tau | {\cal F}_t ]$ is bounded for any ${\mathbb F}$ stopping time $\tau$ and any time $t$ such that $0 \leq t \leq \tau \leq T$ (the bound is uniform in $\tau$), can we draw any conclusion on the uniform integrability of $\{X_\tau\}_{\tau \in {\cal T}}$? Here ${\cal T}$ denotes the family of all stopping times on $[0, T]$.