The refined closed-manifold 3D index conjecture

Let $M$ be a closed $3$-manifold obtained by Dehn filling a $1$-cusped $3$-manifold $N$ with a $1$-efficient ideal triangulation $\mathcal{T}$. Applying the Gang–Yonekura formula defines an index $\mathcal{I}_M(q)$. The refined closed-manifold 3D index conjecture. The index is well-defined and

$$\mathcal{I}_{M}(q)= \begin{cases} 1+P(q),&P(q)\in q^{\frac12}\mathbb{Z}[[q^{\frac12}]]\text{ if }M\text{ is hyperbolic},\\ 0,&\text{if }M\text{ is a lens space},\\ 1,&\text{if }M\text{ is a Seifert fibered space with base orbifold the }(2,3,5), (2,3,4),\text{ or }(2,3,3)\text{-sphere, or a Seifert fibered space with hyperbolic base, or a manifold with Sol geometry},\\ 2,&\text{if }M\text{ is a Seifert fibered space with base orbifold the }(2,2,n)\text{-sphere}. \end{cases}$$

Here $S^3$ and $S^2\times S^1$ are included among lens spaces. The claim refines the earlier closed-manifold pattern attributed to Gang and is motivated by computed examples, including cases without absolute convergence; the source supplies no resolution evidence.