Logical Comparison of CWk with Low-Rank MSO

Determine how the logic $\mathsf{CW}^k$ compares in expressive power with low-rank monadic second-order logic.

Background

The paper introduces $\mathsf{CW}^k$ as a logic characterizing the distinguishing power of the k-dimensional dense Weisfeiler--Leman algorithm. The conclusion asks whether this logic has a known relationship with low-rank MSO, a comparison that could clarify the expressive boundaries of dense homomorphism indistinguishability.