Kapovich–Millson saturation conjecture for simply laced root systems

Let $G$ be a split reductive group over $\x211a$ and let $k_G$ be the positive integer from the saturation theorem: for dominant integral weights $\lambda$, $\mu$, and $\nu$ such that $\lambda^*+\mu+\nu$ lies in the root lattice, positivity of $c_{k\mu,k\nu}^{k\lambda}(G)$ for some $k\in\mathbb{N}$ implies positivity of $c_{k_G\mu,k_G\nu}^{k_G\lambda}(G)$. Kapovich–Millson saturation conjecture. If the root system of $G$ is simply laced, then $k_G$ can be chosen to be $1$. This conjecture predicts the sharp saturation factor for tensor product multiplicities of simply laced reductive groups; the paper's theorem proves the relevant Newell–Littlewood saturation result, but the supplied text does not establish whether this general conjecture has been resolved.