The butterfly permutation parameter conjecture

Let $q=2^n$ with $n$ odd and $\gcd(i,n)=1$. For

$$R_i(x,y)=(x+\alpha y)^{2^i+1}+\beta y^{2^i+1},$$

set

$$V_i(x,y)=(R_i(x,y),R_i(y,x)).$$

Let $\Gamma$ be the parameter set defined in equation. Butterfly permutation parameter conjecture. If $V_i$ is a permutation over $\mathbb{F}_q^2$ with boomerang uniformity $4$, then $(\alpha,\beta)\in\Gamma$. The paper reports computational evidence for the necessity of this condition when $q=2^3$ and $q=2^5$, while the general assertion is left for interested readers to solve.