Finiteness conjecture for characteristic numbers of conic-line arrangements

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

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

The conjecture is motivated by the absence of known families of conic-line arrangements with the number of lines and conics tending to infinity and characteristic number greater than $\frac{5}{2}$. The source describes it as a challenging conjecture and gives no evidence of a resolution.