The simpliciality conjecture for arrangements with fewer than half double points

Let $\mathcal{A}$ be an arrangement consisting of $n$ pseudolines, and let $t^\mathcal{A}_2$ denote its number of double points. Its characteristic polynomial $\chi(\mathcal{A},t)$ need not split over $\mathbb{R}$. Simpliciality conjecture. If

$$t^\mathcal{A}_2<\frac{n}{2},$$

then $\mathcal{A}$ is simplicial. The conjecture is motivated by the fact that the only two known arrangements with fewer than $n/2$ double points are simplicial, while equality can also occur for non-simplicial arrangements.