Analytic continuation of radial pure partition functions

Let $\mathcal{Z}_{\alpha}(\boldsymbol{\theta})$ and $\mathcal{Z}_{\alpha}^{\eta}(\boldsymbol{\theta})$ denote the zero-spin and spin-$\eta$ pure partition functions, respectively, indexed by a radial link pattern $\alpha$. Analytic-continuation conjecture. The functions can be analytically continued to every $\kappa\in(0,8)$.

$$\mathcal{Z}_{\alpha}(\boldsymbol{\theta}),\ \mathcal{Z}_{\alpha}^{\eta}(\boldsymbol{\theta})\quad\text{can be analytically continued to all }\kappa\in(0,8).$$

The conjecture is motivated by the fact that the radial meander matrix is not always invertible for rational $\kappa$; it asserts that the pure partition functions nevertheless extend through those parameter values. The source gives no resolution.