On the pronormality of subgroups of odd index in finite simple symplectic groups

A subgroup $H$ of a group $G$ is pronormal if the subgroups $H$ and $H^g$ are conjugate in $\langle H,H^g\rangle$ for every $g\in G$. It was conjectured in [1] that a subgroup of a finite simple group having odd index is always pronormal. Recently the authors [2] verified this conjecture for all finite simple groups other than $PSL_n(q)$, $PSU_n(q)$, $E_6(q)$ и $^2E_6(q)$, where in all cases $q$ is odd and $n$ is not a power of $2$, and $P\operatorname{Sp}_{2n}(q)$, where $q\equiv\pm3\pmod8$. However in [3] the authors proved that when $q\equiv\pm3\pmod8$ and $n\equiv0\pmod3$, the simple symplectic group $P\operatorname{Sp}_{2n}(q)$ has a nonpronormal subgroup of odd index, thereby refuted the conjecture on pronormality of subgroups of odd index in finite simple groups.
The natural extension of this conjecture is the problem of classifying finite nonabelian simple groups in which every subgroup of odd index is pronormal. In this paper we continue to study this problem for the simple symplectic groups $P\operatorname{Sp}_{2n}(q)$ with $q\equiv\pm3\pmod8$ (if the last condition is not satisfied, then subgroups of odd index are pronormal). We prove that whenever $n$ is not of the form $2^m$ or $2^m(2^{2k}+1)$, this group has a nonpronormal subgroup of odd index. If $n=2^m$, then we show that all subgroups of $P\operatorname{Sp}_{2n}(q)$ of odd index are pronormal. The question of pronormality of subgroups of odd index in $P\operatorname{Sp}_{2n}(q)$ is still open when $n=2^m(2^{2k}+1)$ and $q\equiv\pm3\pmod8$.