The crepant-resolution conjecture for DT4 invariants of $[\mathbb C^4/\mathbb Z_r]$

Let $r$ be a positive integer, let $[[200~\mathbb C^4/\mathbb Z_r]$ be the quotient Calabi–Yau $4$-orbifold considered in the source, and let $q_0,\ldots,q_{r-1}$ be variables. Write

$$q_{[i,j]}:=q_iq_{i+1}\cdots q_j\quad (i\leq j),\qquad Q:=q_{[0,r-1]}.$$

Let $M(a,b):=\prod_{n=1}^{\infty}(1-ab^n)^{-n}$ be the refined MacMahon series and let $\widetilde M(a,b):=M(a,b)M(a^{-1},b)$. The series $\mathsf{DT}_{[\mathbb C^4/\mathbb Z_r],\mathscr O}(m,q_0,\ldots,q_{r-1})$ is defined analogously to the toric case. Cao–Kool–Monavari's crepant-resolution conjecture. There exists a choice of orientation such that

$$\begin{aligned} \mathsf{DT}_{[\mathbb C^4/\mathbb Z_r],\mathscr O}(m,q_0,\ldots,q_{r-1})={}&M(1,-Q)^{-\frac{m}{s_4}\left(\frac{r(s_1+s_2)}{s_3}+\frac{(s_1+s_2)(s_1+s_2+s_3)}{rs_1s_2}\right)}\\\\ &\cdot\prod_{0<i\leq j<r}\widetilde M(q_{[i,j]},-Q)^{-\frac{m(s_1+s_2)}{s_3s_4}}. \end{aligned}$$

This conjecture gives the predicted refined MacMahon product for the DT4 theory of the quotient orbifold and is motivated by the crepant resolution correspondence. Its general status is left open in the supplied text; the surrounding paper studies this orbifold case in the context of toric Calabi–Yau $4$-orbifolds.