The regularized-series conjecture for the barcode kernel

Let $q$ and $\kappa$ be the model parameters, let $\mathcal{F}_n^s(0)$ be the orthogonal functions appearing in the barcode kernel, and let $\mathcal{K}^{\mathrm{barcode}}(s,t)$ be the limiting kernel for $s,t\in\mathbb{Z}$. Regularized barcode-kernel conjecture. For all $s,t\in\mathbb{Z}$ with $s\ge t$, the specified regularized partial sums converge and equal $\mathcal{K}^{\mathrm{barcode}}(s,t)$; using the alternative truncation and regularization factor specified in the statement likewise yields $\mathcal{K}^{\mathrm{barcode}}(s+1,t+1)$. This conjecture proposes a regularization of the otherwise divergent kernel series and identifies it with the scaling limit; its validity is supported numerically and remains open.