APN 3-design conjecture for two quartic-dependent exponents

Let $q=2^n$, with $n\geq5$ odd, and let $B_s$ be the associated base block. Two-exponent 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 in either of the following cases: $s=2^{(n-1)/2}+2^{(n-1)/4}-1$ when $n\equiv1\pmod4$, or $s=2^{(n-1)/2}+2^{(3n-1)/4}-1$ when $n\equiv3\pmod4$. These cases were included among conjectures computationally confirmed for $n\in\{5,7\}$; the general statement remains open.