Cao–Kool's cohomological DT/PT vertex correspondence

Let $\lambda,\mu,\nu,\rho$ be finite plane partitions, with at most two of them non-empty in the stable-pairs case. Let $\mathsf V^{\mathrm{coho},\mathrm{DT}}_{\lambda\mu\nu\rho}(Q)$ and $\mathsf V^{\mathrm{coho},\mathrm{PT}}_{\lambda\mu\nu\rho}(Q)$ be the cohomological DT and PT fourfold vertices. Cao–Kool's conjecture. There are choices of signs such that

$$\mathsf V^{\mathrm{coho},\mathrm{DT}}_{\lambda\mu\nu\rho}(Q)=\mathsf V^{\mathrm{coho},\mathrm{PT}}_{\lambda\mu\nu\rho}(Q)\,\mathsf V^{\mathrm{coho},\mathrm{DT}}_{\varnothing\varnothing\varnothing\varnothing}(Q).$$

This correspondence was established as a cohomological vertex conjecture in the cited work of Cao and Kool and is used here as a limit of the K-theoretic correspondence.