Spoiler--Duplicator Game for CWk

Construct a Spoiler--Duplicator game that characterizes equivalence in the logic $\mathsf{CW}^k$.

Background

Classical Weisfeiler--Leman equivalence admits logical and game-theoretic characterizations. After introducing $\mathsf{CW}^k$ for dense Weisfeiler--Leman, the paper asks whether an analogous Spoiler--Duplicator game exists for this logic.