Helleseth's three-valued conjecture

Let $F$ be a finite field of characteristic $p$, and let $d$ be an invertible exponent over $F$. The Weil sum $W_{F,d}$ is called three-valued when it takes three distinct values.

Helleseth's three-valued conjecture. If $[F:{\mathbb F}_p]$ is a power of $2$, then there is no invertible exponent $d$ over $F$ such that $W_{F,d}$ is three-valued.

The conjecture concerns which finite fields support three-valued Weil sums. It was proved in characteristics $2$ and $3$, while the supplied status evidence indicates that the general conjecture has been resolved.