Quasi-Jacobi form property for Gromov–Witten series of Hilbert schemes of points on K3 surfaces

Let $S$ be a K3 surface, let $S^{[n]}$ be its Hilbert scheme of $n$ points, and let $F^{S^{[n]}}_{g,\ell}(\mathrm{taut};\gamma_1,\ldots,\gamma_N)$ be the corresponding Gromov–Witten generating series. Let $\Delta(q)=q\prod_{n\geq1}(1-q^n)^{24}$ and let $\Gamma_0(\ell)$ be the congruence subgroup of matrices in $\mathrm{SL}_2(\mathbb Z)$ whose lower-left entry is divisible by $\ell$. For bihomogeneous insertions, set $k=n(2g-2+N)+\sum_i\mathsf{wt}(\gamma_i)-10$.

Quasi-Jacobi form property. For all $\ell>0$,

$$F^{S^{[n]}}_{g,\ell}(\mathrm{taut};\gamma_1,\ldots,\gamma_N)\in\frac{1}{\Delta(q)^{\ell}}\operatorname{QJac}_{k+12\ell,\ell(n-1)}(\Gamma_0(\ell)).$$

This would constrain each series to finitely many coefficients and explain its modular and Jacobi transformation behavior. The supplied text does not state whether the property has been proved.