On the Tu-Zeng Permutation Trinomial of Type $(1/4,3/4)$

Let $q$ be a power of $2$. Recently, Tu and Zeng considered trinomials of the form $f(X)=X+aX^{(1/4)q^2(q-1)}+bX^{(3/4)q^2(q-1)}$, where $a,b\in\Bbb F_{q^2}^*$. They proved that $f$ is a permutation polynomial of $\Bbb F_{q^2}$ if $b=a^{2-q}$ and $X^3+X+a^{-1-q}$ has no root in $\Bbb F_q$. In this paper, we show that the above sufficient condition is also necessary.