The exactness conjecture for local coefficient systems on complete regions

Let $\Delta$ be the Bruhat–Tits building of dimension $d$, let $\mathbb{X}$ be the coefficient system, and let $\mathcal{X}$ be a set of facets. For each $i\leq d$, write $C_i(\mathcal{X},\mathbb{X})$ for the space of functions in $C_i(\Delta,\mathbb{X})$ supported in $\mathcal{X}$. The restricted chain complex is

$$0\to C_d(\mathcal{X},\mathbb{X})\to\dots\to C_1(\mathcal{X},\mathbb{X})\to C_0(\mathcal{X},\mathbb{X})\to\mathbb{X}.$$

Assume that either (A) $\mathcal{X}$ is a complete region in $\Delta$, or (B) $\mathcal{X}$ consists of a single face of the hyperspecial chamber $C$. Exactness conjecture. The restricted chain complex is exact.

The source says that exactness is unclear in general but is suspected in several important cases. It later develops results in type $\widetilde{A}_2$ and reformulates the complete-region case, so the conjectural exactness remains open in the supplied text.