Vertex characterization conjecture for key-polynomial Newton polytopes

For compositions $\gamma$ and $\alpha$, write $\gamma\preceq\alpha$ when $\lambda(\gamma)=\lambda(\alpha)$ and $w(\gamma)\leq w(\alpha)$ in Bruhat order, where $\lambda(\gamma)$ is obtained by sorting the parts and $w(\gamma)$ is the shortest permutation sending $\lambda(\gamma)$ to $\gamma$. The paper proves that $\beta\preceq\alpha$ implies that $\beta$ is a vertex of ${\sf Newton}(\kappa_\alpha)$. Vertex characterization conjecture. Conversely, every vertex of ${\sf Newton}(\kappa_\alpha)$ satisfies $\beta\preceq\alpha$. This remains open in the source.