Hyperspaces with the Attouch-Wets topology homeomorphic to $l_2$

It is shown that the hyperspace of all nonempty closed subsets ${\text{Cld}}_{AW}(X)$ of a separable metric space $X$ endowed with the Attouch-Wets topology is homeomorphic to a separable Hilbert space if and only if the completion of $X$ is proper, locally connected and contains no bounded connected component, $X$ is topologically complete and not locally compact at infinity.