The alternating-group soluble-graph diameter conjecture

Let $p\geqslant 11$ be a prime with $p\equiv 3\pmod{4}$. For the alternating group $A_p$, let $\delta_{\mathcal{S}}(A_p)$ denote the diameter of its soluble graph. Alternating-group soluble-graph diameter conjecture.

$$\delta_{\mathcal{S}}(A_p)\geqslant 4.$$

A positive solution would imply that there are infinitely many groups with soluble-graph diameter at least $4$, resolving the open problem posed immediately before it; the source notes that $p\geqslant 11$ is necessary because $\delta_{\mathcal{S}}(A_7)=3$.