One main obstacle for doing Mina's story for general 3-manifold, rather than just $\Sigma \times \R$, is that, the general $U_q(sl_2)$-representation has too many simple objects.
The usual RT / WZW / CS theory, by pass theory problem, by working with integral, rather than
Start with dual nef partitions on $\Delta$ and $\nabla$.
This note is my attempt to understand how the mirror construction works, and how it compares with the monomial-divisor toric correspondence.