BPS-crystal expansion of D6 $qq$-characters

Let $Z=X\times\mathbb{C}$ be a toric Calabi–Yau four-fold, where $X$ is a toric Calabi–Yau three-fold with quiver $Q=(Q_0,Q_1)$ and deformation parameters $\{q_I\}_{I\in Q_1}$. A three-dimensional BPS crystal $\Lambda$ is a set of colored atoms with color map $\mathrm{c}:\Lambda\to Q_0$ and origin atom $\mathfrak{o}$. For an atom $\cube$, define

$$\chi_{X,x}(\cube)=x\prod_{I\in\operatorname{path}[\mathfrak{o}\to\cube]}q_I.$$

Let $\mathsf{A}_i(x)$ and $\mathsf{W}_i(x)$ be the D0- and D6-brane operators, $\mathfrak{q}_i$ the instanton parameters, and $K$ the parameter in the second character. The coefficients $\mathcal{Z}^{\mathrm{D6}}_i[\Lambda^{(i)}]$ and $\mathcal{Z}^{\mathrm{D6}}_i[K,\Lambda^{(i)}]$ are determined by commutativity with screening charges.

D6 $qq$-character BPS-crystal conjecture. For $i\in Q_0$, the $qq$-characters are

$$\mathsf{T}_{i}(x)=\sum_{\Lambda^{(i)}}\mathfrak{q}_{i}^{|\Lambda^{(i)}|}\mathcal{Z}^{\mathrm{D6}}_{i}[\Lambda^{(i)}]:\mathsf{W}_{i}(x)\prod_{\cube\in\Lambda^{(i)}}\mathsf{A}_{\mathrm{c}(\cube)}^{-1}(\chi_{X,x}(\cube)):,$$ $$\mathsf{T}^{K}_{i}(x)=\sum_{\Lambda^{(i)}}\mathfrak{q}_{i}^{|\Lambda^{(i)}|}\mathcal{Z}^{\mathrm{D6}}_{i}[K,\Lambda^{(i)}]:\frac{\mathsf{W}_{i}(x)}{\mathsf{W}_{i}(Kx)}\prod_{\cube\in\Lambda^{(i)}}\mathsf{A}_{\mathrm{c}(\cube)}^{-1}(\chi_{X,x}(\cube)):.$$

They satisfy

$$[\mathsf{T}_{i}(x),\mathscr{Q}_{j}(x')]=0,\qquad [\mathsf{T}^{K}_{i}(x),\mathscr{Q}_{j}(x')]=0.$$

The coefficients are fixed by these commutation relations, and the monomials are classified by three-dimensional BPS crystals. The construction and its coefficients are explicitly presented as conjectural and are left for future clarification.