The block-Toeplitz limit conjecture for the barcode kernel

Let $K^{\mathrm{barcode}}_{(L)}(s,t)$ be the pre-limit barcode kernel for $s,t\in\mathbb{Z}$, and let $q$ be the model parameter. Barcode-kernel limit conjecture. For all $s,t\in\mathbb{Z}$, the limit $\lim_{L\to\infty}\mathbf{i}^{t-s}q^{2L(s-t)}K^{\mathrm{barcode}}_{(L)}(s,t)$ exists and defines $\mathcal{K}^{\mathrm{barcode}}(s,t)$. The limiting kernel is symmetric in $s$ and $t$ and is $2\times2$ block Toeplitz. This conjecture supplies the global limiting kernel underlying the barcode process; it is based on numerical evidence and remains open.