Gruenbaum–Giessen h-vector inequality for simplicial arrangements

Let $\mathcal{A}$ be a simplicial arrangement in $\mathbb{P}^3$, and let $h_i^{\mathcal{A}}$ denote its $h$-vector entries.

Gruenbaum–Giessen h-vector conjecture. One has

$$h_2^{\mathcal{A}} > \sum_{i \geq 3} h_i^{\mathcal{A}}.$$

This is presented as a conjecture from Gruenbaum and Giessen. The source does not state a resolution, so the inequality remains open here.