The indecomposable-root conjecture for simply-laced Weyl groups

Let $G$ be a group of simply-laced type with Weyl group $W$, and let $w\in W$. For $k\in I_w^B$, choose a reduced expression $\mathbf i$ of $w$ satisfying $d_{\mathbf i}(k)=d_w(k)$, and define

$$\check\eta_w(k)=\check\alpha_w^{\mathbf i}(k).$$

Indecomposable-root conjecture. The root $\check\eta_w(k)$ is indecomposable in $\check R^+(w)$. In particular, the set

$$\{\check\eta_w(k)\mid k\in I_w^B\}$$

constructed in this way satisfies the expected properties stated for $\mathcal B_{w,B}$.

The claim is motivated by the construction of bases from rightmost roots and is illustrated by a type $D_5$ example in the source. Its resolution is not established in the supplied material.