Helleseth's vanishing conjecture

Let $F$ be a finite field of characteristic $p$ with $|F|>2$, and let $d$ be an invertible exponent over $F$ satisfying

$$d\equiv 1\pmod{p-1}.$$

The Dedekind determinant is

$$D_{F,d}=\prod_{a\in F^*}W_{F,d}(a).$$

Helleseth's vanishing conjecture. There is some $a\in F^*$ such that

$$W_{F,d}(a)=0.$$

Equivalently, $D_{F,d}=0$. The source reports partial congruence results but no proof of the conjecture in general.