Nonexistence of prime nonsplit two-component links with full symmetry

Let $L$ be a prime, nonsplit two-component link, and let $\Sigma(L)$ denote its intrinsic symmetry group. The link has full symmetry when $\Sigma(L)=\Gamma_2$, where $\Gamma_2$ is the full symmetry group for two-component links. The full-symmetry nonexistence conjecture. There does not exist a prime, nonsplit two-component link $L$ with full symmetry:

$$\Sigma(L)=\Gamma_2.$$

The source notes that no nonsplit examples with full symmetry were known, and that such a link would have linking number zero, nonalternating structure, and components of the same knot type, with that knot type itself possessing full symmetry.