Quantum invariant detection conjecture for the A-ideal

Let $K$ be a knot in $S^3$ whose complement has sufficiently regular $SU(2)$-representation variety. Let $T^2$ be the boundary torus of the knot complement, let $M$ be the $3$-manifold obtained by doubling the knot complement, let $f\in K_{-1}(T^2)$ be a peripheral skein, and let $Z_r(M,f)$ denote the Witten–Reshetikhin–Turaev invariant of $M$ at level $r$. The quantity

$$\limsup_{r\rightarrow\infty}\frac{1}{r}\lvert Z_r(M,f)\rvert$$

defines a seminorm on $K_{-1}(T)$ whose radical is the $A$-ideal of $K$.

Quantum invariant detection conjecture. Under these hypotheses, $\limsup_{r\rightarrow\infty}\frac{1}{r}\lvert Z_r(M,f)\rvert$ defines a seminorm on $K_{-1}(T)$, and its radical is the $A$-ideal of the knot.

This conjecture proposes that asymptotic Witten–Reshetikhin–Turaev invariants of the double of a knot complement detect the $A$-ideal. The source presents it as motivated by the preceding analysis and gives no resolution evidence.