The Harris-Voisin conjecture on special components of the Noether-Lefschetz locus

Let $V$ be the parameter space of smooth complex surfaces of degree $d$ in $\mathbb{P}^{3}$, and let $\operatorname{NL}_d$ be the Noether-Lefschetz locus, whose components of codimension $a=\binom{d-1}{3}$ are called general and whose other components are called special. Harris-Voisin conjecture. The union of all special components of $\operatorname{NL}_d$ is not Zariski dense in $V$. Harris's original finiteness conjecture was disproved by Voisin's counterexamples, while this weaker density statement is the conjecture formulated in the paper.