Higher-rank DT/PT correspondence for the four-fold vertex with leg boundary conditions

Let $\vec{\pi}_a$ be $n$-tuples of plane partitions for the four legs, and let $\vec{v}=(v_1,\ldots,v_n)$ and $\vec{w}=(w_1,\ldots,w_n)$ be the flavor fugacities of the $n$ D8 and D8$'$ branes. Denote the corresponding higher-rank Donaldson–Thomas and Pandharipande–Thomas partition functions by $\mathcal{Z}^{\mathrm{DT}\\,\mathrm{JK}}_{\vec{\pi}_1\vec{\pi}_2\vec{\pi}_3\vec{\pi}_4}$ and $\mathcal{Z}^{\mathrm{PT}\\,\mathrm{JK}}_{\vec{\pi}_1\vec{\pi}_2\vec{\pi}_3\vec{\pi}_4}$. Higher-rank DT/PT correspondence for leg boundary conditions. One has

$$\mathcal{Z}^{\mathrm{DT}\\,\mathrm{JK}}_{\vec{\pi}_1\vec{\pi}_2\vec{\pi}_3\vec{\pi}_4}[\mathfrak{q},q_{1,2,3,4}]=\operatorname{MF}\left[\prod_{\alpha=1}^{n}\frac{w_{\alpha}}{v_{\alpha}}\right]\mathcal{Z}^{\mathrm{PT}\\,\mathrm{JK}}_{\vec{\pi}_1\vec{\pi}_2\vec{\pi}_3\vec{\pi}_4}[\mathfrak{q},q_{1,2,3,4}].$$

This asserts the DT/PT correspondence for arbitrary rank and four-leg plane-partition boundary conditions; its resolution is not supplied in the source.