Tetrahedron-instanton construction with D6-branes

Let $Z=X\times\mathbb{C}$, where $X$ is a toric Calabi–Yau three-fold, with quiver $Q=(Q_0,Q_1)$ and deformation parameters $\{q_I\}_{I\in Q_1}$. Let $\mathsf{A}_i(x)$ and $\mathsf{W}_i(x)$ be the D0- and D6-brane operators, and set

$$\mathsf{A}_{\underline{k}}=\prod_{i\in Q_0}\prod_{I=1}^{k_i}\mathsf{A}_i(x_{i,I}),\qquad \mathsf{W}_{\underline{n}}={:\!\prod_{i\in Q_0}\prod_{\alpha=1}^{n_i}\mathsf{W}_i(v_{i,\alpha})\!:}.$$

Tetrahedron-instanton BPS/CFT conjecture. After suitable conditions, the deformation parameters can be reparametrized to at most three independent parameters. The partition function with multiple D6-branes wrapping $X$ is proportional to

$$\frac{1}{\underline{k}!}\oint\prod_{i\in Q_0}\prod_{I=1}^{k_i}\frac{dx_{i,I}}{2\pi\iota x_{i,I}}\langle\mathsf{A}_{\underline{k}}^{-1}\mathsf{W}_{\underline{n}}\rangle.$$

This realizes the BPS/CFT correspondence, and the poles are labeled by three-dimensional BPS crystals. The statement is presented as a conjectural general construction for tetrahedron instantons.