Symmetry conjecture for three-valued Weil sums

Let $F$ be a finite field, let $d$ be an invertible exponent over $F$, and let $W_{F,d}$ be three-valued. A three-valued Weil sum is called symmetric when its two nonzero values on $F^*$ are opposites of one another.

Symmetry conjecture. $W_{F,d}$ is symmetric.

The source notes that this property is not known for all three-valued Weil sums, although all examples in its table have it.