Jack-parameter interpolation between complex and real origami

Establish that the generating series \(\mathcal{R}_\mu(t,\mathbf{p};b)\) interpolates between the quasimodular generating functions for connected complex origami in \(\mathcal{H}(\mu_1-1,\ldots,\mu_k-1)\) and the quantum modular generating functions for connected real origami in \(\mathcal{H}(\mu_1-1,\mu_1-1,\ldots,\mu_k-1,\mu_k-1)\).

Background

The authors introduce $\mathcal{R}(t,\mathbf{p};b)$ from Jack functions, with specializations at $b=0$ and $b=1$ giving the generating functions for connected complex and real origami, respectively. They note that the complex specialization is quasimodular and that the corresponding real specialization is expected to be quantum modular if the earlier quantum-modularity conjecture holds. The proposed interpolation asks whether the entire Jack-parameter family connects these two modularity phenomena.