The simple-crossing conjecture for complex supersolvable line arrangements

Let $t_2$ denote the number of crossing points of multiplicity $2$ in a complex line arrangement. An arrangement is nontrivial if it is neither a pencil nor a near pencil, and it is supersolvable in the sense used for line arrangements. Simple-crossing conjecture. Every nontrivial complex supersolvable line arrangement has

$$t_2>0.$$

This asserts that such an arrangement must contain a point of multiplicity $2$. The paper presents this as open and notes that no nontrivial supersolvable complex arrangement with $t_2=0$ is known.