We defined the peripheral ideal to be the extension to the noncommutative torus of the kernel of the inclusion map of skein algebra of the torus into the skein module of the complement of the knot. Via an identification of the skein algebra of the torus with the symmetric part of the noncommutative torus the ideal corresponds to the ideal of the imageof the $SL_2\mathbb{C}$-characters of the knot group in the characters of $\mathbb{Z}\times \mathbb{Z}$. We found a way of seeing the colored Jones polynomial of the knot as lying in the dual to the $SL_2\mathbb{C}$-characters of the knot group, and we found thatthe colored Jones polynomial is in the annihilator of the peripheral ideal.
Thang and Stavros stepped back from the picture, and found a formal connection between the Jones polynomial and the noncommutative torus, and then just defined their ideal to be the annihilator of the Jones polynomial. Using formal properties of the $R$-matrix they were able to give an axiomatic proof that their ideal was nontrivial.
The conjecture is about the relation between the formal definition of quantum invariantsand their concrete realization. The Kauffman bracket skein module of a knot complement is a deformation quantization of the unreduced scheme of the $SL_2\mathbb{C}$-characters of its fundamental group. The conjecture that the peripheral ideal is nontrivial is motivated by this idea, and the fact that the $SL_2\mathbb{C}$-character variety of a nontrivial knot, is nontrivial, meaning the $A$-ideal is nontrivial. This should mean that the peripheral ideal is nontrivial.
The orthogonality between the peripheral ideal and the colored Jones polynomial should lead to data about the $SL_2\mathbb{C}$-character variety of the knot being expressed in the aggregate behavior of the colored Jones polynomial of the knot.
