The conformal welding injectivity conjecture

Let a Jordan curve have a conformal welding homeomorphism, and let a curve be conformally removable when every homeomorphism of the Riemann sphere conformal off it is a Möbius transformation. The conformal welding injectivity conjecture. The conformal welding correspondence is injective if and only if the corresponding Jordan curve is conformally removable; equivalently, a Jordan curve is uniquely determined by its welding homeomorphism modulo Möbius transformations if and only if it is conformally removable. This conjecture has been open for over 50 years. It has been claimed by many authors, but earlier proofs were incomplete or incorrect; the paper explains that it would follow from the positive-area conjecture.