Origami interpretation of Jack coefficients for general parameter

Interpret the coefficients \(r_\lambda(b)\) of \(\mathcal{R}(t,\mathbf{p};b)\) in terms of origami counts for values of \(b\) other than the complex and real specializations.

Background

The paper defines $\mathcal{R}(t,\mathbf{p};b)$ using Jack functions and denotes its power-sum coefficients by $r_\lambda(b)$. At $b=0$, the resulting series encodes connected complex-origami counts, while at $b=1$, it encodes connected real-origami counts. The authors explicitly ask whether the intermediate or other parameter values have a combinatorial interpretation as origami enumerations.