Characterization of cluster complexes by root independence and full support

Let $\mathcal{S}$ be the relevant Coxeter-system word set, and let $Q$ be a word in $\mathcal{S}$. For a spherical subword complex $\mathcal{SC}(Q)$, call it root-independent if, for every facet $I$, the multiset

$${{\sf R}}(I)=\big\{\!\big\{ {{\sf r}}(I,i)\mid i\in I\big\}\!\big\}$$

is linearly independent, and call it of full support if every position of $Q$ belongs to some facet. A Coxeter element $c$ and a fixed reduced word $\mathrm{w_\circ}(c)$ determine the word $c\,\mathrm{w_\circ}(c)$.

Cluster-complex characterization conjecture. The following statements are equivalent: (i) up to commutations of consecutive commuting letters, $Q=c\,\mathrm{w_\circ}(c)$ for some Coxeter element $c$; (ii) $\mathcal{SC}(Q)$ is root-independent and of full support.

The conjecture proposes that these two properties characterize cluster complexes among spherical subword complexes. The supplied passage gives no resolution, so the conjecture remains open.