DT/PT orbifold topological vertex correspondence for the cyclic group

Let $G=\mathbb{Z}_{n}$, and let $V^{n}_{\lambda\mu\nu}$ and $W^{n}_{\lambda\mu\nu}$ denote the DT and PT $\mathbb{Z}_{n}$-vertices, respectively. A partition $\nu$ is called multi-regular when it satisfies the multi-regularity condition for the $\mathbb{Z}_{n}$-colored vertex. Define

$$V^{n}_{\emptyset\emptyset\emptyset}=M(1,q)^{n}\prod_{0<a\leq b<n}M(q_{a}\cdots q_{b},q)M(q_{a}^{-1}\cdots q_{b}^{-1},q),$$

where

$$M(v,q)=\prod_{m=1}^{\infty}\frac{1}{(1-vq^{m})^{m}}.$$

DT/PT $\mathbb{Z}_{n}$-orbifold vertex correspondence. If $\nu$ is multi-regular, then

$$V^{n}_{\lambda\mu\nu}=V^{n}_{\emptyset\emptyset\emptyset}W^{n}_{\lambda\mu\nu}.$$

This is the orbifold analogue of the DT/PT topological vertex correspondence, relating the DT and PT generating functions after multiplication by the empty-boundary MacMahon factor. The source states that this conjecture is proved in Section 6.