Equivariant K-theory Schubert coefficient formula via equivariant K-jeu de taquin

Let $\lambda,\mu,\nu$ be partitions indexing equivariant $K$-theory Schubert classes, and let ${\tt EqINC}(\nu/\lambda,|\mu|)$ denote the relevant set of equivariant increasing tableaux. For such a tableau $T$, let ${\tt KErect}(T)$ be its equivariant K-rectification, ${\tt sgn}(T)$ its sign, and ${\tt wt}_K(T)$ its K-theoretic weight. The equivariant K-theory coefficient conjecture. The equivariant $K$-theory Schubert structure coefficient is

$$K_{\lambda,\mu}^{\nu}=\sum_T {\tt sgn}(T)\cdot{\tt wt}_K(T),$$

where the sum is over all $T\in{\tt EqINC}(\nu/\lambda,|\mu|)$ such that ${\tt KErect}(T)=T_\mu$. This gives a tableau formula for the coefficients in equivariant $K$-theory Schubert multiplication, extending the preceding equivariant jeu de taquin construction; its resolution is not indicated in the supplied text.