Permutation-polynomial conjecture for the polynomials $f_{a,q}$

Let $\mathbb F_q$ be the finite field with $q$ elements, let $p=\operatorname{char}\mathbb F_q$, and for $a\geq 1$ define

$$f_{a,q}=X^{q-2}+X^{q^2-2}+\cdots+X^{q^a-2}.$$

A polynomial over $\mathbb F_{q^e}$ is a permutation polynomial if it induces a permutation of $\mathbb F_{q^e}$. Permutation-polynomial conjecture. For $e\geq 2$ and $1\leq a\leq pe-2$, $f_{a,q}$ is a permutation polynomial of $\mathbb F_{q^e}$ if and only if either

$$a=2\quad\text{and}\quad q=2,$$

or

$$a=1\quad\text{and}\quad \operatorname{gcd}(q-2,q^e-1)=1.$$

The conjecture concerns the permutation properties of the polynomials $g_{n,q}$, a class containing $f_{a,q}=g_{q^{a+1}-2,q}$. The paper states that it confirms this conjecture, so the claim is now known in the stated range.