The Knutson–Vakil puzzle conjecture for equivariant Grassmannian K-theory

Let $\Delta_{\lambda,\mu,\nu}$ be the triangular puzzle region whose left, right, and bottom boundary labels are respectively $\lambda$, $\mu$, and $\nu$. A KV-puzzle is a filling of this region by the prescribed puzzle pieces, with the stated rotation, gashing, attachment, and nonlocal placement rules. Its weight $\operatorname{wt}(P)$ is the product of a factor $-1$ for each KV-piece and a factor $1-\frac{t_a}{t_b}$ for each equivariant piece, where $a$ and $b$ are determined by the prescribed diagonal and antidiagonal arrows to the $\nu$-boundary. The Knutson–Vakil puzzle conjecture.

$$K_{\lambda,\mu}^{\nu}=\sum_{P}\operatorname{wt}(P),$$

where the sum is over all KV-puzzles $P$ of $\Delta_{\lambda,\mu,\nu}$. This proposed puzzle rule was intended to compute the equivariant structure constants in the K-theory of Grassmannians, but the paper gives a counterexample, so the conjecture is false.