Ulam reconstruction conjecture for infinite graphs of finite vertex degrees

Let $G$ be an infinite graph with vertex set $E$, and let $\deg(x)$ denote the degree of $x$. Ulam reconstruction conjecture. If

$$\forall x\in E,\quad \deg(x)<\infty,$$

then $G$ is Ulam-reconstructible. The source gives this as a further sufficient condition for reconstruction of infinite graphs, without supplying evidence of a proof or refutation.