The cohomology tensor-product conjecture for the Coxeter complex

Let $G$ be a group with spherical generating data $(S,\mathcal {S})$, let $B$ be the subgroup used to form the Coxeter complex $|\Phi|$, and let $K$ be the associated chamber complex. Write $A_{\mathbf q}$ for the Hecke algebra, $\mathcal {I}(A_{\mathbf q})$ for the coefficient system on $K$, and let $F(G/B)$ be the right $G$-module of finitely supported $\mathbf Q$-valued functions on $G/B$. For each degree $i$, write $H^i_c(|\Phi|)$ for compactly supported cohomology and $H^i(K;\mathcal {I}(A_{\mathbf q}))$ for cohomology with these coefficients.

The cohomology tensor-product conjecture. There is an equality

$$H^i_c(|\Phi|)=H^i(K;\mathcal {I}(A_{\mathbf q}))\otimes_{A_{\mathbf q}}F(G/B).$$

The preceding cochain-level tensor-product identity makes this the natural expected extension to cohomology. The source does not provide evidence resolving the conjecture.