Aganagic–Vafa conjecture relating the augmentation and Q-deformed A-polynomials

Let $K\subset S^3$ be a knot, and let $\mathbf{A}_K(e^x,e^p,Q)$ denote the three-variable $Q$-deformed $A$-polynomial associated with the mirror construction for the resolved conifold. Let $\operatorname{Aug}_K(\lambda,\mu,U)$ be the three-variable augmentation polynomial. Aganagic–Vafa conjecture. For every knot $K$,

$$\mathbf{A}_K(e^x,e^p,Q)=\operatorname{Aug}_K(\lambda=e^x,\mu=e^p,U=Q).$$

The conjecture would connect knot contact homology with the $Q$-deformed $A$-polynomial and, through the latter, with information from topological strings and knot homologies. It has strong circumstantial evidence but was not rigorously proved in the source.