Lately I have been studying in the subject of degree theory, specifically definability results related to $\mathcal{D}$. A famous conjecture in the field due to Slaman and Woodin is that the only automorphism group of $\mathcal{D}$ is the trivial one. However, it seems in the 1999 paper: "Upper cones as automorphism bases" Cooper gives a counterexample which is pointed out in other sources to be wrong. What is the fault of his argument?