Fox's slice-ribbon conjecture

Let $K\subset S^3=\partial D^4$ be a knot. It is slice if it bounds a smoothly, properly embedded disk in $D^4$; it is ribbon if it bounds a disk whose only singularities are ribbon singularities, equivalently if some number of bands can be attached to $K$ to produce an unlink. The slice-ribbon conjecture. Every slice knot is ribbon. Every ribbon knot is already known to be slice, so the conjecture is precisely the converse implication and remains a famous open problem.