Convergence in $H^{-2}$ of $L^2$-functions with limit in $L^2$

Question

Assume a sequence $f_n$ in $L^2(\mathbb{R}^d)$ converges in $H^{-2}$ (w.r.t. its norm topology) to a limit $f \in L^2(\mathbb{R}^d)$. In this case, can one improve the convergence, for instance to weak convergence in $L^2(\mathbb{R}^d)$? Note that I do not explicitly assume the sequence $f_n$ to be bounded in $L^2(\mathbb{R^d})$.

Answer 1

I don't think so. If this was true this would imply boundedness in $L^2$ for your sequence and in the particular case when $f=0$, this would mean that strong convergence in $H^{-2}$ implies boundedness in $L^2$, which is not plausible at all.