APN incidence-structure conjecture for a family of 3-designs

Let $q=2^n$. Let $n\geq5$ be odd, let $s=2^{2i}-2^i+1$, and assume $\gcd(3i,n)=1$. Let $B_s$ be the associated base block and let $\operatorname{GA}_1(q)$ denote the relevant affine group. APN 3-design conjecture. The incidence structure

$$\mathbb{AP}_{n,s}=\left(\operatorname{GF}(q),\operatorname{GA}_1(q)(B_s)\right)$$

is a $3$-$\left(q,\frac q2,\frac{q(q-4)}8\right)$ design. The claim was confirmed by Magma for $n\in\{5,7\}$; its general validity remains open.