The PL sphere-or-ball conjecture for intervals with the NOF property

Let $W$ be a Coxeter group with automorphism $\theta$ having the NOF property, and let $\operatorname{Br}(\iota(\theta))$ be the associated poset. For an interval $[u,v]\subseteq \operatorname{Br}(\iota(\theta))$, call it full when it has the property specified in the paper, and let $(u,v)$ denote its proper part. Let $\rho$ be the rank function.

PL sphere-or-ball conjecture. If $[u,v]$ is full, then $(u,v)$ is a PL sphere of dimension $\rho(v)-\rho(u)-2$. Otherwise, $(u,v)$ is a PL ball of the same dimension.

The claim would follow if the relevant $\mathbb{Z}$-acyclic complexes in the preceding theorem could be shown to be collapsible; the supplied context does not state that this has been achieved.