Suppose that $X_n, n=1,2,\dots,$ is a sequence of random functions in $D([0,\infty), \mathbb{R}^k)$, and let $d$ be the Skorohod metric on $D([0,\infty), \mathbb{R}^k)$. We write $X_n\Rightarrow X$ if $d(X^n, X)\rightarrow 0$ as $n\rightarrow\infty$
There is a general result out there that if $X\in C([0,\infty), \mathbb{R}^k)$, then $X_n\Rightarrow X$ if and only if $\|X_n - X\|_T\rightarrow 0$ for all $T>0$ (I assume almost surely), where for $x\in D([0,\infty), \mathbb{R}^k)$, $$\|x\|_T = \max_{1\le l\le k}\sup_{0\le t\le T}\vert x_l(t)\vert.$$
Suppose, on the contrary, it is known that $\|X_n-X\|_T\Rightarrow 0$ as $n\rightarrow\infty$ for each $T>0$ (here the weak convergence is of the sequence of random variables $\|X_n-X\|_T$, $n=1,2,\dots$, in $\mathbb{R}$). Can one conclude that $X_n\Rightarrow X$ as $n\rightarrow \infty$ (switching back to weak convergence of random processes in $D([0,\infty), \mathbb{R}^k)$)? If not, a counterexample would be appreciated. If so, a reference would be very much appreciated.