The elliptic genus formula for generalized Kummer varieties

Let $A$ be an abelian surface over $\mathbb{C}$, let $A^{[\ell]}$ be its Hilbert scheme of length-$\ell$ subschemes, and let $A^{\langle\ell-1\rangle}$ be the fiber over $0$ of the composition of the Hilbert–Chow morphism with the sum map. Let $\widetilde\phi_{-2,1}(\tau,\nu)$ be the weak Jacobi form of weight $-2$ and index $1$ used in the source, and let $V_\ell$ be the operator defined there. The generalized Kummer elliptic genus conjecture. The elliptic genus of $A^{\langle\ell-1\rangle}$ is given by

$$\mathcal{E}_{A^{\langle\ell-1\rangle}}(\tau,\nu)=\ell^4\frac{\widetilde\phi_{-2,1}\vert_{V_\ell}(\tau,\nu)}{\widetilde\phi_{-2,1}(\tau,\nu)}.$$

This formula proposes a uniform expression for the elliptic genera of generalized Kummer varieties in terms of a Jacobi-form Hecke-type operator. The supplied passage gives no proof or resolution status, so the claim remains open in this database.