For any set $X$ we let $[X]^2 = \big\{\{x, y\}: x\neq y \in X\big\}$. Consider the following statement:

(S) : If $G =(V,E)$ is a simple, undirected graph such that $E \neq [X]^2$, then there is $e^* \in [X]^2 \setminus E$ such that $G \not \cong (V, E\cup\{e^*\})$.

For finite graphs, (S) is true since adding any edge changes the degree sequence. Does (S) hold for infinite graphs as well?