When does convergence in $D[0,\infty)$ imply convergence of functions at infinity?

Question

Suppose we have Càdlàg functions $f_n:[0,\infty)\rightarrow [0,1]$ and we know that $f_n\rightarrow f$ in the Skorokhod topology. When does $\lim_{n\rightarrow\infty} \lim_{x\rightarrow\infty} f_n(x) = \lim_{x\rightarrow \infty} f(x) $?

Are there conditions on the sequence $f_n$ and/or the limit $f$ that guarantee it? For example, does monotonicity of $f$ help?