Combinatorial proof of the Cauchy identity for double Schubert polynomials

Question

The Cauchy identity for double Schubert polynomials states $$ \mathfrak{S}_w(x;-y) = \sum_{\substack{u,v \in S_n \\ w=v^{-1}u \\ l(w) = l(v) + l(u)}} \mathfrak{S}_u(x)\mathfrak{S}_v(y).$$

Is there a combinatorial proof of this identity, akin to the proof of the Cauchy identity for Schur functions via RSK and the dual Cauchy identity via dual RSK? For example, some procedure that takes a pipe dream with $x$ and $y$ weights to a pair of pipe dreams with only $x$ or $y$ weights?

Answer 1

Yes this is proven in the paper RC-graphs and Schubert polynomials, using double rc-graphs (pipe dreams). See section 4 of the link.