The WRT invariant formula in terms of chat Z invariants

Let $M_3$ be a 3-manifold, let $b_1$ denote its first Betti number, and let $a$ and $b$ label (almost) abelian flat connections on $M_3$. Write $\operatorname{CS}(a)$ for the ChernSimons invariant and $\mathcal S_{ab}$ for the corresponding matrix. The WRT--\widehat Z formula. One expects

$$\operatorname{WRT}(M_3,k)=\left(\frac{-i}{\sqrt{2k}}\right)^{1-b_1}\sum_a e^{2\pi i k\operatorname{CS}(a)}\sum_b\mathcal S_{ab}\,\widehat Z_b(q)\bigg|_{q\to e^{\frac{2\pi i}{k}}}.$$

Here the labels $a$ and $b$ run over sets of the same cardinality, though they may be of different nature. The proposed formula would extend the relation between homological $q$-series invariants and Witten--Reshetikhin--Turaev invariants to manifolds with positive first Betti number, using almost abelian flat connections.