K-theory structure constants for minuscule Kac–Moody flag varieties

Let $X=G/P$ be a Kac–Moody flag variety, let $m \in W^P$ be $\Lambda$-minuscule, and let $\mathcal{P}_m$ be the associated $d$-complete poset. For $u,v,w \leq m$ in Bruhat order, let $\lambda,\mu,\nu \subseteq \mathcal{P}_m$ be the order ideals corresponding to $u,v,w \in W^P$, respectively. Structure-constant conjecture. The structure constants satisfy

$$K_{u,v}^w=t_{\lambda,\mu}^\nu.$$

Thus the K-theory multiplication constants for the corresponding interval in the Kac–Moody flag variety agree with those of the combinatorially defined algebra $K(\mathcal{P}_m)$. This proposal builds on the unique-rectification-target conjecture: under that conjecture, $K(\mathcal{P}_m)$ is an associative, commutative, unital algebra whose structure constants alternate transparently in degree, and the equality would identify it with the relevant part of $K(X)$.