Finiteness conjecture for characteristic numbers of complex line arrangements

Let a line arrangement be a finite collection of lines in the complex projective plane, and let $\gamma(\mathcal{L})$ denote its characteristic number. For a real number $\varepsilon>0$, consider line arrangements whose characteristic number exceeds $2.5+\varepsilon$. Finiteness conjecture. For a given $\varepsilon>0$ there exists only finitely many line arrangements $\mathcal{L}$ such that

$$\gamma(\mathcal{L})>2.5+\varepsilon.$$

The conjecture predicts that characteristic numbers of complex line arrangements cannot remain uniformly above $2.5$ along infinitely many arrangements. The source presents it as an extremely challenging conjecture motivated by the scarcity of known families with characteristic numbers in $(\frac{5}{2},\frac{8}{3}]$; no resolution is given.