Helleseth's zero-value conjecture for decimated m-sequence cross correlation

Let $p$ be a prime, let $n$ be a positive integer, and let $d$ be a decimation parameter. Define $\text{S}_d(z)=\text{C}_d(z)+1$, where $z\in\operatorname{GF}(p^n)^*$. Helleseth's conjecture. If $p^n>2$ and

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

then $\text{S}_d(z)=0$ for some $z\in\operatorname{GF}(p^n)^*$. The paper states that this conjecture is confirmed for the setting studied there, while no general resolution is given in the supplied text.