Li–Li–Helleseth–Qu's permutation-inverse bent-function conjecture

Let $q=2^e$ with $e$ an even positive integer, let ${\bb F}_{q^2}$ be the finite field of order $q^2$, and for each $\alpha\in{\bb F}_{q}^*$ define

$$f_{\alpha}(x)=\operatorname{Tr}_{q^2}\bigl(\alpha(\sigma^{-1}(x))^3\bigr),$$

where $\sigma(X)=X+X^d+X^{dq}$ and $d=(q^2+q+1)/3$. Li–Li–Helleseth–Qu's conjecture. The function $f_{\alpha}$ is bent over ${\mathbb F}_{q^2}$ if and only if $\alpha$ is not a cube in ${\mathbb F}_{q}$. This conjecture concerns an explicit bent-function family arising from the inverse of a permutation polynomial; the supplied text presents it as a conjecture proposed in 2023, without indicating that it had been resolved in the source context.