The looped-subgraph energy conjecture for simple graphs

Let $G$ be a simple graph, let $V(G)$ be its vertex set, and for $S\subseteq V(G)$ let $G_S$ denote the graph obtained by adding loops at the vertices in $S$. Write ${\mathcal E}(G)$ for the energy of a graph, namely the sum of the absolute values of the eigenvalues of its adjacency matrix. Looped-subgraph energy conjecture. For every simple graph $G$, there exists $S\subseteq V(G)$ such that

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

The preceding theorem proves the corresponding non-strict inequality for bipartite graphs, while this conjecture asks for a choice of loop set that strictly increases the energy for every simple graph.