For a graph $G$, its energy $\mathcal{E}(G)$ is the sum of absolute values of the eigenvalues of its adjacency matrix, the matching number $μ(G)$ is the number of edges in a maximum matching of $G$, while $Δ$ is the maximum vertex degree of $G$. Akbari, Alazemi and Anđelić in [Appl. Anal. Discrete Math. 15 (2021), 444--459] proved that $\mathcal{E}(G) \leq 2μ(G)$ when $G$ is connected and $Δ\geq6$, and conjectured that the same inequality is also valid when $2\leqΔ\leq5$. Here we first computationally enumerate small counterexamples for this conjecture and then provide two infinite families of counterexamples.