The triple-tangency conjecture for the twice tangent sphere map

Let $K$ be a knot and let $\psi_K^{(2,2)}$ be its twice tangent sphere map, which assigns to a pair of distinct points of $K$ the sphere tangent to $K$ at both points, wherever this sphere is defined. Triple-tangency conjecture. If $K$ is non-trivial, then $\psi_K^{(2,2)}$ is not injective; equivalently, there is a sphere $\Sigma$ tangent to $K$ at three or more points. This predicts a multiple tangency phenomenon for every non-trivial knot, but the source gives no resolution.