Topological invariance of the quantum gluing ideal for knot complements

Let $M$ be a knot complement with a triangulation consisting of tetrahedra $\{\Delta_i\}_{i=1}^N$, and let $\hat{\mathcal I}_M$ be the ideal constructed from this triangulation as defined in the quantum gluing construction. Set

$$a=\frac12,\qquad \kappa_j=1\quad\text{for }j=1,\ldots,N.$$

Topological invariance conjecture. The construction of $\hat{\mathcal I}_M$ produces a topological invariant of $M$, independent of the triangulation and of any other choices made.

The claim asserts invariance under the triangulation changes and auxiliary choices underlying the quantum gluing construction. The supplied passage gives the parameter restrictions needed for the quantum $2$-$3$ Pachner move and for longitude and meridian path deformations, but does not state that the resulting topological invariance has been proved.