Sikora's strong AJ conjecture for knots

Let $K$ be a knot in $S^3$. Let $A_K$ be its recurrence ideal in the quantum torus, let $\sigma$ act by $\sigma(M^kL^l)=M^{-k}L^{-l}$, and let $A_K^{\sigma}$ be the $\sigma$-invariant part. Let $vepsilon$ denote specialization at $t=-1$, and let $\mathfrak p$ be the A-ideal, namely the kernel of the restriction map from the boundary character variety to the character variety of the knot complement.

Strong AJ conjecture. Suppose $K$ is a knot in $S^3$. Then

$$\sqrt{vepsilon(A_K^{\sigma})}=\mathfrak p.$$

This conjecture relates the A-ideal of the knot to the specialized, invariant recurrence ideal. The preceding discussion reports that the ordinary AJ conjecture is known for several knots and knot families, but gives no general resolution of the strong AJ conjecture.