Modularity and topological invariance conjecture for completed numerator series

Let $R$ be a plane curve germ over $\mathbb{C}$, and let $\widehat{\mathit{NZ}}_R(t)$ denote its completed numerator series. Modularity and topological invariance conjecture. There is a formal power series $H_R(t;q)\in\mathbb{Z}[[q,t]]$ such that

$$\widehat{\mathit{NZ}}_R(t)=H_R(t;\mathbb{L}^{-1});$$

for complex $t,q$ with $|q|<1$, $H_R(t;q)$ converges; there is a rational number $\kappa(R)$ such that $q^{\kappa(R)}H_R(1;q)$ and $q^{\kappa(R)}H_R(-1;q)$ are Fourier expansions of weight-zero modular functions; and $H_R(t;q)$, or at least $H_R(\pm1;q)$, is determined by the topology of the link associated to $R$. The convergence assertion is attributed to earlier work for a special case, but the combined conjecture is presented as an open proposal.