The simple $\mathscr{B}_{\psi}$-groups

In a finite group $G$, $\psi(G)$ denotes the sum of element orders of $G$. A finite group $G$ is said to be a $\mathscr{B}_{\psi}$-group if $\psi(H) < |G|$ for any proper subgroup $H$ of $G$. In \cite{Lazorec} Lazorec asked: "what can be said about the $\mathscr{B}_{\psi}$ property of the finite simple groups $\operatorname{PSL}(2, q)$?" In this paper, we answer this question for the case of not only the finite simple groups $\operatorname{PSL}(2, q)$ but also all other finite simple groups. We show that if $S$ is a finite simple group, such that $S \neq Alt(n)$ for any $n \geq 14$, then $S$ is a $\mathscr{B}_{\psi}$-group.