Same-order type characterization conjecture for nonabelian simple groups

Let $S$ be a nonabelian simple $\alpha_n$-group and let $G$ be an $\alpha_n$-group, where an $\alpha_n$-group is a group whose same-order type $\alpha(G)$ has $n$ elements. Assume that

$$|G|=|S|.$$

Same-order type characterization conjecture. Then $G\cong S$.

This asks whether a nonabelian simple group is determined among $\alpha_n$-groups by its order and same-order type. The paper gives this as a conjecture after proving a characterization of $A_5$, but provides no resolution of the broader claim.