The positive-area conjecture for non-removable curves

Let $\textrm{CH}(\gamma)$ be the set of homeomorphisms of the Riemann sphere that are conformal off a curve $c(cgamma)$. A curve is conformally removable if every element of $\textrm{CH}(cgamma)$ is a Möbius transformation. The positive-area conjecture. If $cgamma$ is a non-removable curve, then there exists $cphicinctext{CH}(cgamma)$ such that $cphi(cgamma)$ has positive area. This conjecture has been open for over 30 years and is proved in the paper for the generic class of Jordan curves consisting of flexible curves.