SIN-property characterization of multi-cluster complexes

Let $(W,S)$ be a finite Coxeter system, let $Q$ be a word in $S$ with complete support, and let $\pi\in W$. Write $\delta(Q)$ for the Demazure product of $Q$, and let a word have the SIN-property when it satisfies the property defined in the source. Let $\Delta(Q,\pi)$ denote the corresponding subword complex.

SIN-property characterization conjecture. The subword complex $\Delta(Q,\pi)$ is isomorphic to a multi-cluster complex if and only if $Q$ has the SIN-property and

$$\pi=\delta(Q)=w_\circ.$$

The condition $\pi=\delta(Q)$ is necessary for the subword complex to be a sphere. The source states that it remains to prove $\pi=w_\circ$ and that $Q$ has the SIN-property.