Akbari et al.'s energy-increasing self-loop conjecture

Let $G$ be a simple graph of order $n\geq 2$ with vertex set $V(G)$. For $S\subseteq V(G)$, let $G_S$ be obtained by attaching a self-loop to every vertex in $S$, and let $\mathcal{E}(G)$ and $\mathcal{E}(G_S)$ denote the graph and self-loop graph energies, respectively.

Akbari et al.'s conjecture. There exists a subset $S\subseteq V(G)$ such that

$$\mathcal{E}(G_S)>\mathcal{E}(G).$$

The paper states that this conjecture is confirmed: a vertex-independent set or its complement satisfies the required inequality.