Consider an empirical processes $\{v_T(\theta), T\geq 1 \}$ and assume it is weakly convergent to the stochastic process $\{ v(\theta); \theta \in \Theta\}$., i.e. $$ E^*(f(v_T(\theta)))\rightarrow E(f(v(\theta))) $$ as $T\rightarrow \infty$ $\forall f$ bounded uniformly continuous function in $\Theta$ and $\forall \theta \in \Theta$ where $E^*$ denote the outer expectation.
I know that this implies convergence in distribution $\forall \theta \in \Theta$ as $T\rightarrow \infty$, i.e. $v_T(\theta)=O_p(1)$ $\forall \theta \in \Theta$, i.e. $$ \lim_{T \rightarrow \infty} F_T(a; \theta)=F(a;\theta) $$ $\forall a$ at which $F$ is continuous, where $F_T(\cdot;\theta)$ is the cdf of $v_T(\theta)$ and $F(\cdot;\theta)$ is the cdf of $v(\theta)$.
Question: Does this imply also $$ \sup_{\theta \in \Theta}v_T(\theta)=O_p(1) $$ ?