Automorphisms of the sphere complex of an infinite graph

For a locally finite, connected graph $\mathrm{\Gamma }$, let $\mathrm{Map}(\mathrm{\Gamma })$ denote the group of proper homotopy equivalences of $\mathrm{\Gamma }$ up to proper homotopy. Excluding sporadic cases, we show $\mathrm{Aut}(S(M_{\mathrm{\Gamma }}))\cong \mathrm{Map}(\mathrm{\Gamma })$, where $\mathcal{S}(M_{\mathrm{\Gamma }})$ is the sphere complex of the doubled handlebody $M_{\mathrm{\Gamma }}$ associated to $\mathrm{\Gamma }$. We also construct an exhaustion of $S(M_{\mathrm{\Gamma }})$ by finite strongly rigid sets when $\mathrm{\Gamma }$ has finite rank and finitely many rays, and an appropriate generalization otherwise.