Existence of class-0 non-affine knots with genus different from 1

In $\mathbb{R}P^3$, a knot is called class-$0$ if it has class $0$, and a knot is non-affine if it is not affine. Its genus is the minimum genus of a good surface bounding it. Existence conjecture. There exist class-$0$ non-affine knots whose genus is not equal to $1$. The preceding discussion constructs infinitely many class-$0$ non-affine knots of genus $1$; this conjecture asks whether such knots also occur with a different genus.