Attack or decompose the ABGL-style composition

Determine whether the ABGL-style composition can be handled by another decomposition or a different attack, despite consisting of two monomial layers separated by a Clifford and therefore not directly satisfying the single-monomial-layer hypothesis of the CMC attack theorem.

Background

The paper’s CMC attack applies to unitary families of the form C₂ M C₁, where M is a single monomial layer and the endpoint operations are Clifford unitaries. The authors explain that substituting the strong PFC base into the ABGL-style composition X^{k₁} U X^{k₃} U X^{k₄} produces (X^{k₁}C₂)PF(C₁X^{k₃}C₂)PF(C₁X^{k₄}), with two PF monomial layers separated by a Clifford.

Because the resulting construction does not directly have the single-monomial-layer form required by their theorem, the paper leaves unresolved whether an alternative algebraic decomposition or a different NP-aided attack can analyze or break this composition.