Coulomb-gas representation of radial pure partition functions

Let $\mathrm{LP}(n,m)$ be the set of radial link patterns, let $\mathcal{Z}_{\alpha}(\boldsymbol{\theta})$ be the pure partition functions, let $\mathcal{J}_{\alpha}^{(m,n)}(\boldsymbol{\theta})$ be the corresponding Coulomb-gas integrals, and let $\mathcal{M}_{\kappa}$ be the affine meander matrix. For irrational $\kappa\in(0,8)$, the Coulomb-gas representation conjecture.

$$\mathcal{J}_{\beta}^{(m,n)}(\boldsymbol{\theta})=\sum_{\alpha\in\operatorname{LP}(n,m)}\mathcal{M}_{\kappa}(\alpha,\beta)\mathcal{Z}_{\alpha}(\boldsymbol{\theta}),\qquad \beta\in\mathrm{LP}(n,m).$$

Conversely,

$$\mathcal{Z}_{\beta}(\boldsymbol{\theta})=\sum_{\alpha\in\operatorname{LP}(n,m)}\mathcal{M}_{\kappa}(\alpha,\beta)^{-1}\mathcal{J}_{\alpha}^{(m,n)}(\boldsymbol{\theta}),\qquad \beta\in\mathrm{LP}(n,m).$$

The conjecture proposes an affine-meander-matrix correspondence between pure partition functions and Coulomb-gas integrals in the zero-spin case. The source presents it as a conjecture and gives no resolution.