Connectivity conjecture for intermediate root subsystems

Let $\Delta$ be the root system under consideration, let $I$ index the complete flag $\mathcal{S}_\bullet^1$, and for each $i$ with $1\leq i\leq |I|$ define

$$\Delta_i=\{\alpha\in\Delta\colon h_\alpha t\in\mathcal{S}_i^1\}.$$

The roots $\{\beta_j\}_{j=1}^{i}$ form the simple roots of $\Delta_i$ under the induced polarization. Connectivity conjecture. For any $1\leq i\leq |I|$, the root subsystem $\Delta_i$ is irreducible, or equivalently, the Dynkin subdiagram formed by $\{\beta_j\}_{j=1}^{i}$ is connected. This conjecture concerns the nested intermediate root systems induced by the complete flag and is used as a crucial property in the analysis of increasing chains; its resolution is not specified in the source.