Radial limit conjecture for GPPV invariants of rational homology spheres

Let $M$ be a rational homology sphere, meaning a closed oriented $3$-manifold with $H_1(M,\mathbb{Q})=0$. Let $\operatorname{Spin}^c(M)$ denote its set of $\operatorname{Spin}^c$ structures, with the conjugation action of $\{\pm1\}$. For a positive integer $k$, let $Z_k(M)\in\mathbb{C}$ be the normalized $\operatorname{SU}(2)$ Witten–Reshetikhin–Turaev invariant, and let $\operatorname{lk}\colon H_1(M,\mathbb{Z})\otimes H_1(M,\mathbb{Z})\to\mathbb{Q}/\mathbb{Z}$ be the linking form. Define $\zeta_k=e^{2\pi i/k}$, $\bm{e}(z)=e^{2\pi i z}$, and, for $a,b\in\operatorname{Spin}^c(M)/\{\pm1\}$, let $\operatorname{Stab}_{\{\pm1\}}(a)$ be the stabilizer of $a$ under the conjugation action and set

$$S_{ab}=\frac{e^{2\pi i k\operatorname{lk}(a,b)}+e^{-2\pi i k\operatorname{lk}(a,b)}}{|\operatorname{Stab}_{\{\pm1\}}(a)|\,|\operatorname{Stab}_{\{\pm1\}}(b)|\sqrt{|H_1(M,\mathbb{Z})|}}.$$

Radial limit conjecture. For each $b\in\operatorname{Spin}^c(M)/\{\pm1\}$, there exist topological invariants $\Delta_b\in\mathbb{Q}$, $c\in\mathbb{Z}_{\ge0}$, and $\widehat{Z}_b(q;M)\in2^{-c}q^{-\Delta_b}\mathbb{Z}[[q]]$ such that $\widehat{Z}_b(q;M)$ converges for $|q|<1$ and, for infinitely many $k\in\mathbb{Z}_{>0}$,

$$Z_k(M)=\lim_{q\to\zeta_k}\frac{1}{\zeta_{2k}-\zeta_{2k}^{-1}}\sum_{a,b\in\operatorname{Spin}^c(M)/\{\pm1\}}\bm{e}\left(k\operatorname{lk}(a,a)\right)S_{ab}\widehat{Z}_b(q;M).$$

The conjecture seeks $q$-series invariants whose radial limits recover the WRT invariants and would provide a route toward categorification and asymptotic expansions for general rational homology spheres. Such constructions are known in important cases, including the Poincaré homology sphere and various Seifert or negative-definite plumbed manifolds, but the asserted existence for every rational homology sphere remains open.