Affine meander-matrix conjecture for pure partition functions

Fix irrational $\kappa\in(0,8)$ and let $\mathrm{LP}(n,m)$ be the set of link patterns. Let $\mathcal{Z}_{\alpha}$ be the pure partition function indexed by $\alpha\in\mathrm{LP}(n,m)$, and let $\mathcal{J}^{(m,n)}_{\alpha}$ be the corresponding Coulomb gas integral. Let $\mathcal{M}_{\kappa}$ denote the affine meander matrix. Affine meander-matrix conjecture. For every $\beta\in\mathrm{LP}(n,m)$,

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

and conversely,

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

The relations identify the Coulomb gas integrals with the pure partition functions through the affine meander matrix; the supplied text gives no resolution or further evidence of their status.