Lam's alternating-sign conjecture for K-theoretic k-Schur functions

Let $n\geq 2$, set $k=n-1$, and let $\Lambda_{(n)}$ be the subalgebra of symmetric functions with basis $g^{(k)}_\lambda$, indexed by partitions $\lambda$. Write $s^{(k)}_\lambda$ for the $k$-Schur function, and define coproduct structure constants by

$$\Delta(g^{(k)}_\lambda)=\sum_{\mu,\nu}c^{\mu\nu}_\lambda g^{(k)}_\mu\otimes g^{(k)}_\nu.$$

Lam's conjecture. The following hold: each $g^{(k)}_\lambda$ is a finite sum of $k$-Schur functions with positive integer coefficients; $(-1)^{|\lambda|-|\nu|-|\mu|}c^{\mu\nu}_\lambda\in\mathbb Z_{\geq0}$, with $c^{\mu\nu}_\lambda=0$ unless $|\mu|+|\nu|\leq|\lambda|$; and, in the expansion $g^{(k)}_\lambda=\sum_\mu a^\mu_\lambda g^{(k+1)}_\mu$, one has $(-1)^{|\lambda|-|\mu|}a^\mu_\lambda\in\mathbb Z_{\geq0}$. The top homogeneous component of $g^{(k)}_\lambda$ is $s^{(k)}_\lambda$. Computational checks in the paper verify the first and third assertions in bounded cases and provide data for the second.