Primitive quasi-Jacobi form property and holomorphic anomaly equation

Let $S\to B$ be an elliptic K3 surface, let $S^{[n]}$ be the Hilbert scheme of $n$ points, and let $F^{S^{[n]}}_g$ denote the generating series in primitive curve classes. For $\mathsf{wt}$-homogeneous classes $\gamma_i$, define $k=n(2g-2+N)+2+\sum_i\mathsf{wt}(\gamma_i)$; the classes, tautological insertions, operator $U$, and operators $T_{e_a}$ are as in the source.

Primitive conjecture.

$$F^{S^{[n]}}_g(\mathrm{taut};\gamma_1,\ldots,\gamma_N)\in\frac1{\Delta(q)}\operatorname{QJac}_{k,n-1},$$

and, assuming this membership, its $G_2$-derivative equals the genus-reduction term, the splitting term, the insertion correction, and the quadratic $T_{e_a}$ term exactly as displayed in the source.