Identify the relationship between the n-strand and Grassmannian splicing embeddings

Determine whether, under the stated identifications of the skew shaped positroid varieties with n-strand braid varieties, the open embedding obtained from the length-additive decomposition of the n-strand braid coincides with the embedding induced by the splicing construction for skew shaped positroids.

Background

The paper realizes a skew shaped positroid both as a braid variety on k strands and, via its open Richardson variety description, as a braid variety on n strands. For a cut at the a-th column, the authors obtain a length-additive decomposition of the relevant n-strand braid into left and right factors. Results cited from forthcoming work then provide an open embedding of the product of the braid varieties associated with these factors into the original braid variety, with the image described by freezing certain cluster variables.

The paper also constructs a splicing embedding directly for skew shaped positroids through the maps associated with the column cut. Although both constructions arise from corresponding decompositions and identifications, the authors explicitly leave unresolved whether the two embeddings agree after translating between the braid-variety and skew-shaped-positroid descriptions.