  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 $\mu(G)$ is the
number of edges in a maximum matching of $G$, while $\Delta$ is the maximum
vertex degree of $G$. Akbari, Alazemi and An{\dj}eli\'c in [Appl. Anal.
Discrete Math. 15 (2021), 444--459] proved that $\mathcal{E}(G) \leq 2\mu(G)$
when $G$ is connected and $\Delta\geq6$, and conjectured that the same
inequality is also valid when $2\leq\Delta\leq5$. Here we first computationally
enumerate small counterexamples for this conjecture and then provide two
infinite families of counterexamples.
