  A subgroup $H$ of a group $G$ is said to be {pronormal} in $G$ if $H$ and
$H^g$ are conjugate in $\langle H, H^g \rangle$ for every $g \in G$. Some
problems in finite group theory, combinatorics, and permutation group theory
were solved in terms of pronormality. In 2012, E. Vdovin and the third author
conjectured that the subgroups of odd index are pronormal in finite simple
groups. In this paper we disprove their conjecture and discuss a recent
progress in the classification of finite simple groups in which the subgroups
of odd index are pronormal.
