A symplectic refinement of shifted Hecke insertion

Buch, Kresch, Shimozono, Tamvakis, and Yong defined Hecke insertion to formulate a combinatorial rule for the expansion of the stable Grothendieck polynomials $G_\pi$ indexed by permutations in the basis of stable Grothendieck polynomials $G_\lambda$ indexed by partitions. Patrias and Pylyavskyy introduced a shifted analogue of Hecke insertion whose natural domain is the set of maximal chains in a weak order on orbit closures of the orthogonal group acting on the complete flag variety. We construct a generalization of shifted Hecke insertion for maximal chains in an analogous weak order on orbit closures of the symplectic group. As an application, we identify a combinatorial rule for the expansion of "orthogonal" and "symplectic" shifted analogues of $G_\pi$ in Ikeda and Naruse's basis of $K$-theoretic Schur $P$-functions.