At most one positive knot per smooth concordance class

Show that each smooth concordance class of knots in S^3 contains at most one positive knot; equivalently, ascertain that no two distinct positive knots are smoothly concordant.

Background

Stoimenow conjectured finiteness properties for positive knots within concordance classes, and later work proved that each concordance class contains only finitely many positive knots. Strengthening this, the conjecture stated here posits uniqueness of a positive representative in any smooth concordance class.

This conjecture is motivated by the minimality of positive knots under ribbon concordance and by Gordon’s uniqueness question for ribbon-concordance minimal representatives.