Countable length everywhere club uniformization

Assume $\mathsf {ZF} + \mathsf {AD}$ and all sets of reals are Suslin. Let $\Gamma $ be a pointclass closed under $\wedge $ , $\vee $ , $\forall ^{\mathbb {R}}$ , continuous substitution, and has the scale property. Let $\kappa = \delta (\Gamma )$ be the supremum of the length of prewellorderings on $\mathbb {R}$ which belong to $\Delta = \Gamma \cap \check \Gamma $ . Let $\mathsf {club}$ denote the collection of club subsets of $\kappa $ . Then the countable length everywhere club uniformization holds for $\kappa $ : For every relation $R \subseteq {}^{