$\newcommand\R{\mathbb R}$Let $(f_n)$ be a sequence in $L^1(\R)$ converging weakly to some $f\in L^1(\R)$. Let $(a_n)$ be sequence in $\R$ converging to $0$. For each natural $n$, let $g_n$ be the $a_n$-shift of $f_n$, so that $g_n(x)=f_n(x-a_n)$ for all real $x$.

Does it then always follow that the sequence $(g_n)$ converges weakly to $f$?

This question is a modification/generalization of the previous question, now deleted by that post's author. I think the question is interesting; at least, I would like to see an answer to it.