The DT/PT crepant resolution conjecture for abelian quotient fourfolds

Let $G=\mathbb{Z}_r<\mathrm{SU}(2)<\mathrm{SU}(4)$ or $G=\mathbb{Z}_2\times\mathbb{Z}_2<\mathrm{SO}(3)<\mathrm{SU}(4)$. Let $X\to\mathbb{C}^4/G$ be the Nakamura $G$-Hilbert crepant resolution and let $\mathcal{X}=[\mathbb{C}^4/G]$. If $R_0,\ldots,R_{|G|-1}$ are the irreducible representations and $\beta_i$ is the curve class corresponding to $R_i$ under Reid's generalized McKay correspondence, then there exist orientations such that

$$\mathcal{Z}^{\operatorname{DT}}_{\mathcal{X},\mathcal{O}_{\mathcal{X}}}(y,q_0,\ldots,q_{|G|-1})=\mathcal{Z}^{\operatorname{DT}}_{X,\mathcal{O}_X}(y,0,q)\mathcal{Z}^{\operatorname{PT}}_{X,\mathcal{O}_X}(y,Q,q)\mathcal{Z}^{\operatorname{PT}}_{X,\mathcal{O}_X}(y,Q^{-1},q),$$

under $Q^{\beta_i}=q_i$ for $i=1,\ldots,|G|-1$ and $q=q_0\cdots q_{|G|-1}$. The DT/PT crepant resolution conjecture. The identity above should hold with these orientations and the specified change of variables. This conjecture is motivated by the three-dimensional crepant resolution correspondence and the DT/PT correspondence for Calabi–Yau fourfolds; it is verified in the paper only in the finite-order cases stated there and remains open generally.