Gusein-Zade, Luengo, and Melle-Hernández's product formula for diagonal $[?]$-equivariant Hilbert schemes

Let the cyclic group $\mathbb{Z}_3$ act diagonally on $\mathbb{C}^2$, and let $b_k(-)$ denote the $k$th topological Betti number. The equivariant Hilbert scheme $[\mathbb{C}^2/\mathbb{Z}_3]^{[n]}$ parametrizes $\mathbb{Z}_3$-invariant zero-dimensional subschemes of length $n$. Gusein-Zade, Luengo, and Melle-Hernández's conjecture. The generating series of its even Betti numbers is

$$1+\sum_{n>0\atop k\geqslant 0}b_{2k}\left(\left[\mathbb{C}^2/\mathbb{Z}_3\right]^{[n]}\right)t^kq^n=\prod_{m\geqslant 1}\frac{1}{1-t^{m-1}q^{3m-2}}\frac{1}{1-t^mq^{3m-1}}\frac{1}{1-t^{m-1}q^{3m}}.$$

This conjectural product formula was introduced in the study of the topology of equivariant Hilbert schemes; the paper proves the highest Betti-number consequences but does not establish the full generating-series identity.