The von Neumann neighborhood conjecture for graph codes

Let $A$ be the adjacency matrix of a graph $\Gamma$ on $n$ vertices with vertex set $V$. Let $C$ be the binary linear code generated by $[I_n|A]$, and let $d(C)$ denote its minimum distance. For a nonempty subset $S$ of $V$, let $\operatorname{von}(S)$ denote its von Neumann neighborhood. The von Neumann neighborhood conjecture. If

$$d(C)=|S|+|\operatorname{von}(S)|,$$

then either $S=\operatorname{von}(S)$ or $S\cap\operatorname{von}(S)=\varnothing$. The observation preceding this conjecture verifies the claim for complete graphs, while its validity for general graphs is left open.