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For an uncountable regular cardinal $\kappa$, choose cofinal sequences $c_{\alpha }:\omega \to \alpha$ for the stationary set of ordinals of cofinality $\omega$. Above any bound, some fixed coordinate exceeds the bound on a stationary subset; Fodor lemma makes that coordinate constant on a stationary subset. Thus the stationary constant fibers, over all coordinates, have unboundedly many values. Regularity makes one coordinate have $\kappa$ such values. Its fibers are disjoint stationary sets; adding all leftover ordinals to one piece partitions $\kappa$.

Start with positions whose entire extension cylinder set lies in the winning open set. At each successor stage add I-positions with some successor already present and II-positions with every successor already present; take unions at limit ordinals. The stable set is the least fixed point of this order-preserving operation. The first entry ordinal supplies a strictly decreasing rank until the winning prefix is reached.