Tu–Zeng–Li–Helleseth conjecture on permutation trinomials over quadratic finite fields

Let $q=2^m$ and let $\alpha,\beta\in\mathbb{F}_{q^2}^*$, with

$$f_{\alpha,\beta}(x)=x+\alpha x^{q(q-1)+1}+\beta x^{2(q-1)+1}.$$

The polynomial $f_{\alpha,\beta}(x)$ is a permutation polynomial of $\mathbb{F}_{q^2}$ if and only if one of the following conditions holds: (1) $\beta=\alpha^{q-1}$ and $\operatorname{Tr}\left(1+\frac{1}{\alpha^{q+1}}\right)=0$; or (2) $\beta(1+\alpha^{q+1}+\beta^{q+1})+\alpha^{2q}=0$, $\beta^{q+1}\neq 1$, and $\operatorname{Tr}\left(\frac{\beta^{q+1}}{\alpha^{q+1}}\right)=0$.

Tu–Zeng–Li–Helleseth conjecture. If $f_{\alpha,\beta}(x)$ permutes $\mathbb{F}_{q^2}$, then condition (1) or condition (2) holds.

This conjecture characterizes the permutation trinomials in this class over $\mathbb{F}_{q^2}$. The source paper states that it was proved there, so the conjecture is resolved.