Knot-decorated WRT decomposition conjecture

Let $M_3$ be a closed oriented $3$-manifold, let $K\subset M_3$ be a knot colored by a representation $\mathcal R$ of $G$, and let $a,b$ range over $\operatorname{Tor} H_1(M_3,\mathbb Z)/\mathbb Z_2$. Let $\Gamma_{K,\mathcal R}$ be the corresponding line operator, let $\widehat Z_b(q;\Gamma_{K,\mathcal R})$ be the associated knot-decorated homological block, and let $S_{ab}$ be the S-matrix from the WRT decomposition conjecture.

Knot-decorated decomposition conjecture. The WRT invariant satisfies

$$Z_{SU(2)_k}[M_3;K,\mathcal R]=(i\sqrt{2k})^{b_1(M_3)-1}\sum_{a,b\in\operatorname{Tor} H_1(M_3,\mathbb Z)/\mathbb Z_2}e^{2\pi i k\ell k(a,a)}S_{ab}\,\widehat Z_b(q;\Gamma_{K,\mathcal R})\big|_{q\rightarrow e^{\frac{2\pi i}{k}}}.$$

The decorated blocks are required to satisfy

$$\widehat Z_b(q;\Gamma_{K,\mathcal R})\in2^{-c}q^{\Delta_b}\mathbb Z[[q]],\qquad \Delta_b\in\mathbb Q,\quad c\in\mathbb Z_+,$$

and to converge for $|q|<1$. This conjecture extends the proposed WRT categorification from bare $3$-manifolds to knots colored by representations, with the same S-matrix. The source reports verification in examples but does not state a general proof.