A degree-12 counterexample to Saniga et al.'s Section 4.4 line-subgeometry conjecture

We refute the line-subgeometry conjecture in Section 4.4, p. 20 of Saniga et al., Hexagons govern three-qubit contextuality, arXiv:2312.07738v3 (Quantum 9 (2025), 1601). The 75-line geometry contained in any three-qubit perp hyperplane has contextuality degree 12 but meets every classically embedded split Cayley hexagon in 15 lines. A 19-line configuration has degree 1 and intersection at least 3. Linear sections and signed Clifford transport prove that every perp in n >= 2 qubits has four times the degree of the full (n-1)-qubit line geometry; every four-qubit perp consequently has degree 252.