The AJ conjecture for the colored Jones polynomial

Let $K$ be a knot. Its colored Jones polynomial is $J_K(n)=J_n(K;q)$ and satisfies a nontrivial recurrence

$$\sum_{j=0}^{d}c_j(q,q^n)J_K(n+j)=0,$$

where $c_j(q,q^n)\in\mathbb{Z}[q,q^n]$. Define operators $E$ and $Q$ by

$$(EJ_K)(n)=J_K(n+1),\qquad (QJ_K)(n)=q^nJ_K(n).$$

The $A_q$-polynomial $A_q(K)(E,Q)$ is the most essential annihilating polynomial of the colored Jones polynomial, and the $A$-polynomial is denoted by $A_K(l,\alpha)$. AJ conjecture. For any knot $K$, the $A$-polynomial $A_K(l,\alpha)$ is equal to $\varepsilon A_q(K)(l,\alpha^2)$ up to multiplication by an element of $\mathbb{Q}(\alpha)$, where $\varepsilon$ is evaluation at $q=1$. This conjecture asserts that the $A_q$-polynomial specializes to the classical $A$-polynomial, making precise the relationship between the colored Jones polynomial and the character variety of the knot complement. No resolution status is supplied here.