The alternating-group derangement bound conjecture

Let $G=A_n$ be a finite simple transitive permutation group with point stabiliser $H$, and let $\delta(G)$ denote the proportion of derangements in the action.

Alternating-group derangement bound conjecture. The following hold:

$$\delta(G)\geqslant\frac{13}{45},$$

with equality if and only if $G=A_8$ and $H={\rm AGL}_3(2)$; and, if $n\geqslant9$,

$$\delta(G)\geqslant\frac{[n!/e]}{n!}-\frac{(-1)^n(n-1)}{n!},$$

with equality if and only if $H=A_{n-1}$.

The second assertion gives the proposed non-asymptotic formula for the natural action and would make $13/45$ the optimal lower-bound constant for alternating groups. The source presents these statements as a conjecture based on computations.