Xie–Li–Wang–Zeng's cyclotomic bent-function conjecture

Let $q=2^e$ with $e$ an even positive integer, let $u$ be a generator of $\mu_{q+1}$, and for $\alpha\in{\mathbb F}_{q}^*$ define $g_\alpha$ on ${\mathbb F}_{q^2}$ by $g_\alpha(0)=0$ and

$$g_\alpha(x)=\operatorname{Tr}_{q^2}\left(\alpha\,\frac{u^{6i}}{(1+u^{2i}+u^{-2i})^3}x^3\right)$$

for $x\in u^i{\mathbb F}_{q}^*$ and $0\leq i\leq q$. Xie–Li–Wang–Zeng's conjecture. The function $g_\alpha$ is bent if and only if $\alpha$ is not a cube in ${\mathbb F}_{q}$. The source states that this is a cyclotomic reformulation of the permutation-inverse family and that the supplied paper proves the corresponding assertion, but the candidate itself is presented as the conjecture being reformulated.