The root-order conjecture for decomposable inversion roots

Let $W$ be a simply-laced Coxeter group, let $w\in W$, and let $\check R^+(w)$ be its associated inversion set. For a reduced expression $\mathbf i$ of $w$, write $<_{\mathbf i}$ for the corresponding order on $\check R^+(w)$. A root $\check\alpha\in\check R^+(w)$ is decomposable if it can be written as a sum of two roots in the relevant root system.

Root-order conjecture. If $\check\alpha\in\check R^+(w)$ is decomposable, then there exists a decomposition

$$\check\alpha=\check\mu+\check\mu'$$

such that, for two different reduced expressions $\mathbf i,\mathbf i'$ of $w$,

$$\check\mu<_{\mathbf i}\check\mu'\qquad\text{and}\qquad \check\mu'<_{\mathbf i'}\check\mu.$$

The statement need not hold for every decomposition of $\check\alpha$, only for at least one. It is motivated by the behavior of indecomposable elements of inversion sets and is presented as an open conjecture.