Integrable hierarchies associated with generalized Schur-series powers

Determine whether the series \(\tau_c(\mathbf{p})=\sum_\lambda d_\lambda^c s_\lambda(\mathbf{p})\) is related to an integrable hierarchy for every integer \(c\).

Background

The authors recall that the special cases $c=-1$ and $c=0$ yield tau functions of the KP and large BKP hierarchies, respectively. They then define the family $\tau_c(\mathbf{p})$ for general integer $c$; the case $c=1$ is connected with the generating function for complex origami. The unresolved question is whether analogous integrable-hierarchy structures persist for other integer exponents.