The almost simple group conjecture for the divisibility of Sylow subgroup numbers

Let $p$ be a prime, and for a finite group $G$ let $u_p(G)$ denote the number of Sylow $p$-subgroups of $G$. Say that $G$ satisfies $\mathbf{DivSyl}(p)$ if, for every subgroup $H\leqslant G$, the integer $\nu_p(H)$ divides $\nu_p(G)$. Let $S$ be a simple group.

Almost simple group conjecture. If $S$ satisfies $\mathbf{DivSyl}(p)$, then every group $L$ satisfying

$$S\leqslant L\leqslant \operatorname{Aut}(S)$$

also satisfies $\mathbf{DivSyl}(p)$.

This would extend the known result for almost simple groups whose socle has order not divisible by $p$. The source presents the assertion as natural because the authors do not know an example of an almost simple group satisfying $\mathbf{DivSyl}(p)$ while its simple socle does not; its resolution is not supplied here.