Integral resolution conjecture for Steinberg representations

Let $W$ be a finite-rank free $\mathbb{Z}$-module, and let $\operatorname{St}(W)$ be the top reduced homology of its Tits building. For $n\geq 3$, define the integral complex with terms

$$\mathbf{S}_i(\mathbb{Z}^{n+1})=\bigoplus_{W_1\oplus\cdots\oplus W_{i+2}=\mathbb{Z}^{n+1}}\operatorname{St}(W_1)\otimes\cdots\otimes\operatorname{St}(W_{i+2})$$

for $-1\leq i\leq n-1$, with $\mathbf{S}_{-1}(\mathbb{Z}^{n+1})=\operatorname{St}(\mathbb{Z}^{n+1})$. $b$-integral resolution conjecture. For $b\geq1$, set $b'=\min(b,\lfloor n/2\rfloor)$. Then the portion

$$\mathbf{S}_{b'}(\mathbb{Z}^{n+1})\longrightarrow\mathbf{S}_{b'-1}(\mathbb{Z}^{n+1})\longrightarrow\cdots\longrightarrow\mathbf{S}_0(\mathbb{Z}^{n+1})\longrightarrow\mathbf{S}_{-1}(\mathbb{Z}^{n+1})\longrightarrow0$$

of the integral resolution is exact. This precise conjecture quantifies the preceding expectation that increasingly long portions of the integral complex become exact as $n$ grows, and the stated bound is used in the paper’s vanishing theorem.