Anzis–Tohăneanu's simple-crossing conjecture for complex supersolvable arrangements

Let $t_2$ denote the number of crossing points of multiplicity $2$ in a complex line arrangement, and let $s$ be the number of its lines. A non-pencil complex supersolvable line arrangement is a supersolvable arrangement that is not a pencil. Anzis–Tohăneanu's conjecture. Every non-pencil complex supersolvable line arrangement of $s$ lines has

$$t_2\geq s/2.$$

This is described as a much stronger conjecture than the assertion that $t_2>0$ for every nontrivial complex supersolvable arrangement, and it remains open in the source.