Descent of preorderlike topoi under free group actions

Let $\mathcal{X}$ be a topos which is preorderlike, meaning that $\mathsf{Topoi}(\mathcal{E},\mathcal{X})$ is a preorder for every topos $\mathcal{E}$. Let $\mathcal{G}$ be a group object in $\mathsf{Topoi}$, and suppose there is a free group action

$$\mu: \mathcal{X} \times \mathcal{G} \to \mathcal{X}.$$

Descent conjecture. The descent object $\mathcal{X}/\mathcal{G}$ of $\mathcal{G}$-equivariant sheaves on $\mathcal{X}$ is also preorderlike.

This proposes that preorderlikeness is preserved by descent along free group actions. The source presents the claim as something that “seems to be true,” so its status is unresolved in the supplied text.