Grünbaum–Shephard conjecture for simplicial arrangements

Prove or disprove the Grünbaum–Shephard conjecture that every irreducible simplicial hyperplane arrangement in real projective three-space has more ordinary intersection lines than non-ordinary intersection lines, equivalently that GS(A) = t₂^(1) − Σ_{q≥3} t_q^(1) is strictly positive.

Background

The Grünbaum–Shephard defect is defined as GS(A) = t₂^1 − Σ_{q≥3} t_q^1, where t_q^1 counts intersection lines contained in exactly q planes. Strict positivity means that more than half of the intersection lines are ordinary, namely contained in exactly two planes.

All seven arrangements analyzed in the paper satisfy the strict inequality. Nevertheless, the authors state that the conjecture remains unresolved in general for irreducible simplicial hyperplane arrangements in three-space.