Mock Jacobi correspondence for dual \widehat{Z}-series of four-fibered Seifert homology spheres

Let $\Phi_p^{[j]}(q)^\vee$ be the vector of mock modular forms dual under $q \to 1/q$ to the linear combination of false theta functions $\Phi^{[j]}_p(q)$ associated to the $\widehat{Z}$ invariants of $\Sigma(p_1,p_2,p_3,p_4)$, where $p=p_1p_2p_3p_4$. Let $h_{(p,r_j)}(q)$ be the coefficients of the theta decomposition for the special mock Jacobi form $\mathcal Q_p$. Mock Jacobi correspondence. The following relation holds at the level of $q$-series:

$$\Phi_p^{[j]}(q)^\vee=C\,h_{(p,r_j)}(q),$$

where $C\in\mathbb{Q}$ is a normalization constant. This conjecture proposes that the dual $q$-series arising from the $\widehat{Z}$ invariants are proportional to the theta-decomposition coefficients of special mock Jacobi forms, extending the correspondence observed in the four-fibered examples and relating Chern--Simons topological invariants to quarter-BPS state-counting structures.