Pairwise non-isomorphism conjecture for Kassami APN 3-designs

Let $n\geq5$ be odd, and let

$$I_n=\left\{i:1\leq i\leq\frac{n-1}{2},\ \gcd(i,n)=1\right\}.$$

Let $\phi(n)$ denote Euler's totient function, and let $\mathbb{KA}_{n,i}$ be the corresponding Kassami APN $3$-designs. Kassami-design non-isomorphism conjecture. The $\frac{\phi(n)}2$ designs $\mathbb{KA}_{n,i}$ are pairwise non-isomorphic. Moreover, they are not equivalent to any designs from APN power functions presented in the paper or to any designs from o-monomials introduced by Ding and Tang. The assertion was computationally confirmed for $n\in\{5,7\}$; the general classification remains open.