The OD-characterizability conjecture for alternating groups

Let $H$ be a finite group, and let $h_{\rm OD}(H)$ denote the number of isomorphism classes of groups $G$ such that $|G|=|H|$ and ${\rm D}(G)={\rm D}(H)$. A group is OD-characterizable when $h_{\rm OD}(H)=1$. The OD-characterizability conjecture for alternating groups. Every alternating group $A_m$ with $m\neq 10$ is OD-characterizable. The conjecture is explicitly proposed as a continuation of known results for alternating groups, including all $A_m$ with $m<100$ except $A_{10}$. It is refuted: the paper states that $A_{125}$ satisfies $h_{\rm OD}(A_{125})\geqslant 3$, and proves that infinitely many alternating groups are not OD-characterizable.