The deletion double-point conjecture for free line arrangements

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

Deletion double-point conjecture. If ${\mathcal A}\setminus\{H\}$ is not free, then

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

The source presents this as a weaker version of the preceding nonfree line-arrangement conjecture. No resolution is supplied in the given text.