The sum-of-element-orders criterion for supersolvability

Let $G$ be a finite group, and let $\psi'(G)$ denote the normalized sum of the element orders of $G$.

Supersolvability conjecture. If

$$\psi'(G)>\frac{31}{77},$$

then $G$ is supersolvable. Moreover, $\psi'(G)=\frac{31}{77}$ if and only if

$$G\cong A_4\times C_m\quad\text{with }(6,m)=1.$$

This conjecture proposes a sharp threshold for supersolvability in terms of the normalized sum of element orders, with $A_4\times C_m$ attaining the boundary value. The supplied text gives no evidence that the conjecture has been resolved.