The barcode-process universality conjecture in the waterfall region

Let $(\mathsf{t},\mathsf{x})$ be a macroscopic point in the waterfall region $\mathcal{W}$, and let $q$ and $\kappa$ be the model parameters. Let $\mathcal{K}^{\mathrm{barcode}}(s,t)$ be the symmetric kernel described by the limiting regularized series, and let $\mathsf{N}<\mathsf{T}$ denote the condition that the waterfall phase is present. Barcode-process universality conjecture. Around any macroscopic point $(\mathsf{t},\mathsf{x})\in\mathcal{W}$, the random configuration of square lozenges on a horizontal slice converges to a determinantal barcode process on $\mathbb{Z}$ with correlation kernel $\mathcal{K}^{\mathrm{barcode}}(s,t)$. This process depends only on $q$ and $\kappa$, is $2\times2$ block Toeplitz, satisfies $\mathcal{K}^{\mathrm{barcode}}(0,0)=1-\mathcal{K}^{\mathrm{barcode}}(1,1)$, has global density $1/2$, and is invariant under even but not odd shifts. The conjecture describes universal local statistics throughout the waterfall phase, independently of the hexagon geometry and macroscopic location; the existence and properties of the limiting kernel are supported by numerical and simulation evidence but are not established rigorously.