Suppose that we have real-valued cadlag stochastic processes $X_n=(X_n(t))_{t\in\mathbb{R}}$ (for $n\in\mathbb{N}$), and $X=(X(t))_{t\in\mathbb{R}}$, such that as $n\rightarrow\infty$
$\quad X_n$ converges weakly to $X$ in the standard Skorohod topology.
Suppose further that we have a sequence of real-valued random variables, $T_n$ (for $n\in\mathbb{N}$), such that as $n\rightarrow\infty$
$\quad T_n$ converges weakly to $0$ in $\mathbb{R}$.
Then is it the case that $X_n(T_n)$ converges weakly to $X(0)$ in $\mathbb{R}$?
Important edit: We also assume that $\mathbb{P}[X\text{ is continuous at } 0]=1$. Otherwise we could find a trivial counter example such as
$X_n(t)=X(t)=\begin{cases}0\quad;\quad t<0\\1\quad;\quad t\geq0\end{cases}\quad$
and
$T_n=-1/n$.