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

Let $\vec{\pi}_{a,+}$ be $n$-tuples of plane partitions and $\vec{\pi}_{a,-}$ be $m$-tuples of plane partitions for each of the four legs. 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 $\mathcal{Z}^{+}_{\vec{\pi}_{1,\pm}\vec{\pi}_{2,\pm}\vec{\pi}_{3,\pm}\vec{\pi}_{4,\pm}}$ and $\mathcal{Z}^{-}_{\vec{\pi}_{1,\pm}\vec{\pi}_{2,\pm}\vec{\pi}_{3,\pm}\vec{\pi}_{4,\pm}}$ the mixed $(\mathrm{DT}|\mathrm{PT})$ and $(\mathrm{PT}|\mathrm{DT})$ vertices. Mixed-vertex DT/PT correspondence. One has

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

Here the mixed vertices combine PT4 counting on the positive brane part with DT4 counting on the opposite brane part. The identity is stated conjecturally and no resolution is given.