Convolution convergence conjecture for affine-Grassmannian clasps

Let $\mathcal{K}_{\mathrm{Gr},m}$ be the geometric categorical $\mathfrak{sl}_\infty$ 2-representation, let $\mathcal{K}^{\pm}_{\mathrm{Gr},m}$ denote its bounded-sided homotopy categories, let $\mathsf{T}_i$ be the braid-group 1-morphism, let $\1_{\lambda}$ be the identity 1-morphism at weight $\lambda$, and let $C$ denote convolution. Let ${\sf{P}}^{\mp}$ be the corresponding clasp. Convolution convergence conjecture. The limit

$$\lim_{n\to\infty}C(\mathsf{T}_i)^{\pm 2n}\1_{\lambda}$$

exists as a 1-morphism in $\mathcal{K}^{\pm}_{\mathrm{Gr},m}$ and is isomorphic to a convolution of ${\sf{P}}^{\mp}$. The preceding lemma proves the first nontrivial convergence case under a specific weight hypothesis; the conjecture asks for the general convergence and identification with a clasp convolution.