Harvey–Moore modularity conjecture for fiberwise rational curves

Let $X$ be a tamed $K3$-fibered Calabi–Yau threefold, and let $n_C$ denote the virtual number of rational curves in the fiberwise class $C\in H_2(X,\mathbf Z)_f$. Suppose that the fiberwise intersection lattice

$$\mathbf M\cong H^2(X,\mathbf Z)_f$$

is unimodular. Write $C^2=2\delta-2$ for the self-intersection number along the fiber, and set $n_\delta=n_C$ for classes with this self-intersection. Harvey–Moore's modularity conjecture. The number $n_C$ depends on $C$ only through $C^2=2\delta-2$, and the generating function

$$\sum_{g\geq 0}n_\delta q^\delta$$

is the $q$-expansion of an explicitly constructed modular form. Harvey and Moore's prediction comes from type $IIA$–heterotic duality and relates curve counts on $K3$-fibered Calabi–Yau threefolds to Borcherds-type automorphic products. The source gives no resolution of this conjecture; its status is therefore open.