Jack-polynomial interpolation conjecture for real and complex origamis

Let $b:=\alpha-1$, and define

$$\mathcal{R}(t,\mathbf{p};b):=\log\left(\sum_n\sum_{\lambda\vdash n}J^{(b)}_\lambda t^n\right)=\sum_n\sum_{\lambda\vdash n}r_\lambda(b)p_\lambda t^n.$$

For $\mu=[\mu_1,\ldots,\mu_k]$, let $\mathcal{R}_\mu(t,\mathbf{p};0)$ be the generating function for connected complex origamis in $\mathcal{H}(\mu_1-1,\ldots,\mu_k-1)$, and let $\mathcal{R}_\mu(t,\mathbf{p};1)$ be the generating function for connected real origamis in $\mathcal{H}(\mu_1-1,\mu_1-1,\ldots,\mu_k-1,\mu_k-1)$.

Jack-polynomial interpolation conjecture. The generating series $\mathcal{R}_\mu(t,\mathbf{p};b)$ interpolates between the quasimodular forms given by the generating functions for connected complex origamis in $\mathcal{H}(\mu_1-1,\ldots,\mu_k-1)$ and the quantum modular forms given by the generating functions for connected real origamis in $\mathcal{H}(\mu_1-1,\mu_1-1,\ldots,\mu_k-1,\mu_k-1)$.

At $b=0$ and $b=1$, the series specializes respectively to the complex- and real-origami generating functions; the complex specialization is known to be quasimodular, while the real specialization is quantum modular if the first conjecture holds. The interpolation claim is presented as an open question.