The false-mock conjecture for orientation-reversed three-manifolds

Let $M_3$ be a three-manifold such that

$$\hat Z_b(M_3;\tau)=q^c\left(\widetilde\vartheta(\tau)+p(\tau)\right),$$

where $\tau\in\mathbb H$, $c\in\mathbb Q$, $\widetilde\vartheta$ is the Eichler integral of a weight-$3/2$ unary theta function $\vartheta$, and $p$ is a polynomial in $q=e^{2\pi i\tau}$. Let $f$ be a weight-$1/2$ weakly holomorphic mixed mock modular form, and let $\vartheta_i$ be theta functions. False-mock conjecture. The orientation-reversed invariant satisfies

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

and the completion of $f$ is

$$\widehat f=f-\vartheta^\ast-\sum_{i\in I}g_i\vartheta_i^\ast,$$

where $I$ is finite and each $g_i$ is a modular function for a discrete subgroup of ${\rm SL}_2(\mathbb Z)$ that either vanishes or has an exponential singularity at every cusp. The conjecture predicts a mock-modular description of orientation-reversed $\hat Z$-invariants, with the holomorphic and non-holomorphic Eichler integrals arising from the same cusp form; the paper presents it as an inspiration and a generalization of earlier versions.