Slaman–Woodin conjecture on the complexity of the c.e. orbit relation

Let $(W_i)_{i\in\mathbb{N}}$ be an effective enumeration of the computably enumerable sets, and write $W_i\approx W_j$ when the two sets lie in the same orbit under automorphisms of the lattice of computably enumerable sets modulo finite sets. Slaman–Woodin conjecture. The index relation

$$\{\langle i,j\rangle: W_i\approx W_j\}$$

is $\Sigma^1_1$-complete. This conjecture concerns the descriptive-set-theoretic complexity of deciding whether two computably enumerable sets are automorphic. The source says that it was made by Ted Slaman and Hugh Woodin in 1989, but supplies no resolution.