Pellegrini–Shumyatsky centralizer-coset conjecture

Let $G$ be a finite non-abelian simple group, let $t\in G$ be an involution, and let $\mathcal{Z}_G(t)$ denote the centralizer of $t$. A coset of the centralizer is a set $g\mathcal{Z}_G(t)$ for some $g\in G$. Pellegrini–Shumyatsky's centralizer-coset conjecture. Each coset of $\mathcal{Z}_G(t)$ contains an element of odd order, unless $G=\operatorname{PSL}(n,2)$ for $n\geq 4$. The conjecture is refuted: the paper gives counterexamples in the alternating groups $A_{8n}$ for all $n\geq 2$; these groups are not among the stated exceptions.