The generalized-quiver partition-function conjecture for HOMFLY-PT homology

Let $K$ be a knot with conormal $L_K$ in the resolved conifold. Suppose the associated symmetric generalized quiver has nodes $1,\dots,m$, multiplicities $\mu_i$, adjacency matrix $C_{ij}$, disk homology classes $\mu_i\log x+a_i\log a$, and invariant self-linkings $q_i$. Define the node variables by

$$x_i=x^{\mu_i}q^{q_i-C_{ii}\mu_i}a^{a_i}(-t)^{C_{ii}\mu_i}.$$

Generalized-quiver partition-function conjecture. After this substitution, the generalized-quiver partition function equals the generating function of Gromov-Witten invariants counting generalized holomorphic curves, equivalently the generating function of symmetrically colored HOMFLY-PT polynomials of $K$, and the refined generalized-quiver partition function equals the generating function of Poincaré polynomials of HOMFLY-PT homology in symmetric representations. This is the paper's main conjectural correspondence between generalized quivers, holomorphic-curve counts, and HOMFLY-PT homology; the source gives no resolution evidence.