Weak convergence in the Skorohod space $D([0, T])$ $\forall T$ implies Weak convergence in $D([0, \infty))$?

Weak convergence in the Skorohod space $D([0, T])$ $\forall T$ implies
Weak convergence in $D([0, \infty))$?

Assume that $Z_n$ are random variables taking value in the Skorohod space
$D([0, \infty),Y)$ (endowed with its usual Skorohod topology) of
right-continuous functions $[0, \infty) \to Y$, where $Y$ is a real
separable Banach space. Assume that $\forall T>0$, $Z_n$ converge weakly
(in distribution) to $Z$ in the space $D([0, T],Y)$, where $Z$ is a
constant random variable defined on $D([0, \infty),Y)$. Does it hold that
$Z_n$ converge weakly to $Z$ in the space $D([0, \infty),Y)$?