Braid varieties are a large class of flag configuration spaces that include (open) Richardson varieties. In 2022, Casals et al. showed that the coordinate rings of these varieties carry a cluster structure by using weaves, a diagrammatic calculus created in connection to symplectic knot theory. In 2025, motivated by results in link homology, Gorsky et al. constructed splicing maps identifying certain principal open subsets of a braid variety with a direct product of smaller braid varieties and conjectured these maps behave well with regards to the cluster structure. In this talk, we explain these results and propose a weave construction that helps study the cluster structure behavior under the splicing map.