Conjecture on disjoint positive root sets in folded cluster structures

Let $D^{\perp}\simeq\mathbf{D}_{2m}$ with $m\geq 2$, let $1\leq l\leq m-1$, and let $\mathcal{P}$ be the power set of $\{1,\cdots,l\}$. For each $\varpi=\{i_{1},\cdots,i_{h}\}\in\mathcal{P}$, let ${\bf s}_{\varpi}$ be the seed obtained by performing the reflections $r_{\alpha_{i_{1}}}\circ\cdots\circ r_{\alpha_{i_{h}}}$, and let $\overline{\Delta}_{{\bf s}_{\varpi},\Pi}^{+}$ denote the associated positive folded root set. Disjointness conjecture. For any two distinct elements $\varpi,\varpi^{'}$ in $\mathcal{P}$, $\overline{\Delta}_{{\bf s}_{\varpi},\Pi}^{+}$ and $\overline{\Delta}_{{\bf s}_{\varpi^{'}},\Pi}^{+}$ have trivial intersection. This predicts that the positive folded root sets arising from the different reflection sequences are pairwise disjoint, providing a potentially interesting application of the folding procedure in positive non-acyclic cases.