Type C $K$-Stanley symmetric functions and Kra\'skiewicz-Hecke insertion

We study Type C $K$-Stanley symmetric functions, which are $K$-theoretic extensions of the Type C Stanley symmetric functions. They are indexed by signed permutations and can be used to enumerate reduced words via their expansion into Schur $Q$-functions, which are indexed by strict partitions. A combinatorial description of the Schur $Q$- coefficients is given by Kra\'skiewicz insertion. Similarly, their $K$-Stanley analogues are conjectured to expand positively into $GQ$'s, which are $K$-theory representatives for the Lagrangian Grassmannian introduced by Ikeda and Naruse also indexed by strict partitions. We introduce a $K$-theoretic analogue of Kra\'skiewicz insertion, which can be used to enumerate 0-Hecke expressions for signed permutations and gives a conjectural combinatorial rule for computing this $GQ$ expansion. We show the Type C $K$-Stanleys for certain fully commutative signed permutations are skew $GQ$'s. Combined with a Pfaffian formula of Anderson's, this allows us to prove Lewis and Marberg's conjecture that $GQ$'s of (skew) rectangle shape are $GQ$'s of trapezoid shape. Combined with our previous conjecture, this also gives an explicit combinatorial description of the skew $GQ$ expansion into $GQ$'s. As a consequence, we obtain a conjecture for the product of two $GQ$ functions where one has trapezoid shape.