Descent conjecture for the first steps of the Shimura curve tower at 3

Let $X_0(\mathfrak{p}_3)$ and $X_0(\mathfrak{p}_3^2)$ be the indicated Shimura curves with models over $\mathbb{F}_{2^3}$. Descent conjecture. The canonical branched cover

$$X_0(\mathfrak{p}_3^2)_{\mathbb{F}_{2^3}}\longrightarrow X_0(\mathfrak{p}_3)_{\mathbb{F}_{2^3}}$$

and the Atkin–Lehner involution on $X_0(\mathfrak{p}_3^2)_{\mathbb{F}_{2^3}}$ descend over $\mathbb{F}_2$. This is the remaining descent assertion needed, together with recursive reduction, to obtain the proposed dense family over $\mathbb{F}_2$ and the value $A'_6(2)=(2^3-1)/6$; no proof is supplied.