  Let $\Bbb F_q$ be the finite field with $q$ elements and let
$p=\text{char}\,\Bbb F_q$. It was conjectured that for integers $e\ge 2$ and
$1\le a\le pe-2$, the polynomial $X^{q-2}+X^{q^2-2}+\cdots+X^{q^a-2}$ is a
permutation polynomial of $\Bbb F_{q^e}$ if and only if (i) $a=2$ and $q=2$, or
(ii) $a=1$ and $\text{gcd}(q-2,q^e-1)=1$. In the present paper we confirm this
conjecture.
