The nonfree line-arrangement double-point conjecture

Let ${\mathcal A}$ be a projective line arrangement in ${\mathbf P}^2_{\mathbb K}$, where ${\mathbb K}$ is a field of characteristic zero. Write $n_2({\mathcal A})$ for the number of double points of ${\mathcal A}$, and call ${\mathcal A}$ free when its associated arrangement is free.

Nonfree line-arrangement double-point conjecture. If ${\mathcal A}$ is not free, then

$$n_2({\mathcal A})>0.$$

The conjecture asks whether every nonfree arrangement over a characteristic-zero field has a double point. The supplied text gives experimental motivation but no proof or resolution.