Lam's alternating-sign conjecture for affine stable Grothendieck polynomials

Let $n\geq2$, set $k=n-1$, and let $G_w$ be an affine stable Grothendieck polynomial for $w\in W_{\mathrm{af}}$. Let $G^{(k)}_\lambda$ and $F^{(k)}_\mu$ denote the corresponding partition-indexed affine stable Grothendieck and affine Schur functions. Define coefficients by

$$G^{(k)}_\mu G^{(k)}_\nu=\sum_\lambda c^{\mu\nu}_\lambda G^{(k)}_\lambda.$$

Lam's conjecture. Every $G_w$ is a finite alternating linear combination of the $G^{(k)}_\lambda$; every $G^{(k)}_\lambda$ is an alternating integer linear combination of the $F^{(k)}_\mu$; $(-1)^{|\lambda|-|\mu|-|\nu|}c^{\mu\nu}_\lambda\in\mathbb Z_{\geq0}$, with $c^{\mu\nu}_\lambda=0$ unless $|\mu|+|\nu|\leq|\lambda|$; and, if $G^{(k+1)}_\mu=\sum_\lambda a^\mu_\lambda G^{(k)}_\lambda$, then $(-1)^{|\lambda|-|\mu|}a^\mu_\lambda\in\mathbb Z_{\geq0}$. The conjecture extends expected positivity and alternating-sign behavior in affine K-theoretic Schubert calculus; the paper presents these as conjectural properties rather than proving them.