Analytic continuation conjecture for the quantum modularity cocycle

Let \gamma=\begin{pmatrix}a&b\c&d\end{pmatrix}\in\operatorname{SL}_2(\mathbb Z) with $c\neq0$, and let $W_\gamma$ be defined on $\mathbb Q\smallsetminus\{\gamma^{-1}(\infty)\}$. Analytic continuation conjecture. The function $W_\gamma$ extends to a real-analytic function on $\mathbb R\smallsetminus\{\gamma^{-1}(\infty)\}$, and its restriction to each component of that set extends holomorphically to the cut plane consisting of the component's half-line together with $\mathbb C\smallsetminus\mathbb R$. This would give a strong analytic form of quantum modularity for the cocycle. The source reports numerical evidence but does not state a proof.