Purdy-type inequality for irreducible simplicial arrangements

Establish whether every irreducible simplicial arrangement of projective planes in real projective three-space satisfies the Purdy-type inequality p(A) − ℓ(A) + n + 2 ≥ 0, where n is the number of planes, ℓ(A) is the number of rank-two flats, and p(A) is the number of rank-three flats.

Background

The paper distinguishes the rank-flat difference G(A) = p(A) − ℓ(A) from the Purdy defect Δ(A) = p(A) − ℓ(A) + n + 2. Although the rank-flat inequality p(A) ≥ ℓ(A) fails for all arrangements examined in the paper, the Purdy defect is nonnegative for each of them.

These computations motivate asking whether the weaker Purdy-type inequality holds universally for irreducible simplicial arrangements in projective three-space.