Support conjecture for key polynomials

Let $\alpha$ be a composition, let $t_{ij}$ swap positions $i$ and $j$, and let $m_{ij}(\alpha)=\alpha+e_i-e_j$. Define $\beta<_\kappa\alpha$ when $\beta$ can be generated from $\alpha$ by a sequence of moves $t_{ij}$ with $\alpha_i<\alpha_j$ and moves $m_{ij}$ with $\alpha_j-\alpha_i>1$. Let $\kappa_\alpha$ be the key polynomial. Key support conjecture.

$$\kappa_\alpha=x^\alpha+\sum_{\beta<_\kappa\alpha}{\sf Key}_{\alpha,\beta}x^\beta,$$

with ${\sf Key}_{\alpha,\beta}>0$ for every $\beta<_\kappa\alpha$. The source gives no proof or resolution of this stronger support description.