The Suslin-cardinal unreachable successor conjecture

Assume ${\sf{AD}}^+$. Let $\kappa$ be a regular Suslin cardinal, and let ${\mathbf{\Gamma}}$ be the pointclass of $\kappa$-Suslin sets. Suslin-cardinal unreachable successor conjecture. Then $\kappa^+$ is ${\mathbf{\Gamma}}$-unreachable. The authors expect that their methods will generalize to settle this conjecture, which is presented as an open problem beyond the results proved in the paper.