The mock modularity conjecture for quantum invariants of orientation-reversed three-manifolds

Let $M_3$ be a three-manifold whose quantum invariants have the form

$$\widehat Z_a(M_3;\tau)=q^c\left(\widetilde g(\tau)+p(\tau)\right),$$

where $c\in\mathbb{Q}$, $\widetilde g(\tau)$ is the Eichler integral of a weight-$\frac{3}{2}$ theta function $g(\tau)$, and $p(\tau)$ is a polynomial in $q$. Mock modularity conjecture. The quantum invariants of the orientation reversal satisfy

$$\widehat Z_a(-M_3;\tau)=q^{-c}\left(f(\tau)+p(-\tau)\right),$$

where $f(\tau)$ is a weight-$\frac{1}{2}$ mock modular form whose shadow is $g(\tau)$. This proposes a relation between quantum invariants of plumbed three-manifolds and mock modular forms; the source notes that outside the weakly negative plumbed setting it can also serve as a definition, while physical quantum invariants exist more generally and allow independent checks.