A result on the sum of element orders of a finite group

Let G be a finite group and ψ(G)=∑g∈Go(g)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (G)=\sum _{g\in {G}}{o(g)}$$\end{document}. There are some results about the relation between ψ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (G)$$\end{document} and the structure of G. For instance, it is proved that if G is a group of order n and ψ(G)>2111617ψ(Cn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi (G)>\dfrac{211}{1617}\psi (C_n)$$\end{document}, then G is solvable. Herzog et al. in (J Algebra 511:215–226, 2018) put forward the following conjecture: