The Dirac–Motzkin conjecture for real line arrangements

Let $t_2$ denote the number of crossing points of multiplicity $2$ in a real line arrangement, and let $s$ be the number of its lines. Dirac–Motzkin conjecture. The inequality

$$t_2\geq \lfloor s/2\rfloor$$

holds for every non-pencil real line arrangement of $s$ lines. This strengthens the elementary lower bound $t_2\geq 3$ for non-pencil real arrangements and remains open in the stated generality.