DT/PT correspondence for mixed four-fold vertices with surface boundary conditions

Let $\vec{\lambda}_{A,+}$ be $n$-tuples of plane partitions and $\vec{\lambda}_{A,-}$ be $m$-tuples of plane partitions for the six surfaces. Let $v_\alpha=e^{\mathfrak{a}_\alpha}$, $w_\alpha=e^{\mathfrak{b}_\alpha}$, $x_\beta=e^{\mathfrak{c}_\beta}$, and $y_\beta=e^{\mathfrak{d}_\beta}$. Denote by $\mathcal{Z}^{+}_{\\{\vec{\lambda}_{A,\pm}\\}}$ and $\mathcal{Z}^{-}_{\\{\vec{\lambda}_{A,\pm}\\}}$ the mixed $(\mathrm{DT}|\mathrm{PT})$ and $(\mathrm{PT}|\mathrm{DT})$ vertices. Mixed surface-vertex DT/PT correspondence. One has

$$\mathcal{Z}^{+}_{\\{\vec{\lambda}_{A,\pm}\\}}[\mathfrak{q},q_{1,2,3,4}]=\operatorname{MF}\left[\prod_{\alpha=1}^{n}\frac{w_{\alpha}}{v_{\alpha}}\prod_{\beta=1}^{m}\frac{y_{\beta}}{x_{\beta}}\right]\mathcal{Z}^{-}_{\\{\vec{\lambda}_{A,\pm}\\}}[\mathfrak{q},q_{1,2,3,4}].$$

The poles from the D8–D8$'$ part give PT4 counting and those from the opposite brane part give DT4 counting. The source states the correspondence conjecturally without giving a resolution.