Unreachability below $\bf{\Delta^1_3}$

Let $\bf{\Gamma}$ be a pointclass, and let $\kappa$ be $\bf{\Gamma}$-reachable when there is a sequence of distinct $\bf{\Gamma}$-sets of length $\kappa$; otherwise, $\kappa$ is $\bf{\Gamma}$-unreachable. Write $\bf{\Gamma}<_w\bf{\Delta^1_3}$ for the weak ordering of pointclasses used in the source. Unreachability conjecture. There is no pointclass $\bf{\Gamma}<_w\bf{\Delta^1_3}$ such that $\aleph_{\omega+1}$ is $\bf{\Gamma}$-reachable. The question is part of the paper's open problems on unreachability; analogous questions at higher projective levels are also left open.