The average-degree energy lower-bound conjecture for non-singular graphs

Let $G$ be a non-singular graph of order $n$, with average degree $\overline{d}$ and energy $\mathcal{E}(G)$ equal to the sum of the absolute values of the eigenvalues of its adjacency matrix. Let $P_4$ denote the path of order $4$, and let $H$ denote the graph shown in the source figure.

Average-degree energy conjecture. Then

$$\mathcal{E}(G)\geq n-1+\overline{d},$$

except for $P_4$ and $H$.

This is presented as a generalization of the preceding energy lower-bound conjecture. The supplied text gives no resolution status; the exceptional graph $H$ is specified only by the source figure.