Dobbertin–Helleseth–Kumar–Martinsen conjecture on a three-valued family of Weil sums

Let $F$ be a finite field of order $q=3^n$, where $n$ is odd and $n>1$, and let $d=3^r+2$ satisfy

$$4r\equiv 1\pmod n.$$

For $a\in F$, define the Weil sum

$$W_{F,d}(a)=\sum_{x\in F}\psi(x^d-ax),$$

where $\psi\colon F\to\mathbb C$ is the canonical additive character. Dobbertin–Helleseth–Kumar–Martinsen conjecture. The Weil sum $W_{F,d}$ is three-valued, with

$$W_{F,d}(a)=\begin{cases}0&\text{for $q-q/3-1$ values of $a\in F^{\times}$,}\\+\sqrt{3q}&\text{for $(q+\sqrt{3q})/6$ values of $a\in F^{\times}$, and}\\-\sqrt{3q}&\text{for $(q-\sqrt{3q})/6$ values of $a\in F^{\times}$.}\end{cases}$$

The conjecture describes a tenth infinite family of three-valued Weil sums; the paper proves it, so the asserted value distribution is established for all parameters satisfying the stated hypotheses.