Harris's topological and analytic finiteness conjectures for special components

Fix an integer $d\geq 5$. Let $\operatorname{NL}_d$ be the Noether–Lefschetz locus, and let a component be special when its codimension is strictly less than the maximal value $\binom{d-1}{3}$. Components may be considered either only as topological irreducible components, ignoring their natural analytic scheme structures as Hodge loci, or as irreducible components with those analytic scheme structures. Harris's conjectures. (1) Ignoring the natural analytic scheme structure, $\operatorname{NL}_d$ has only finitely many topological special components. (2) With the Hodge-locus analytic scheme structure, $\operatorname{NL}_d$ has only finitely many special irreducible components. These conjectures concern the finiteness of special Noether–Lefschetz components. The supplied status evidence says that questions about the largest $d'$ for which the conjectures hold for all $d\leq d'$ remain open, so the conjectures are recorded as open.