Cohomological dimension and boundary cohomology conjecture for relatively hyperbolic pairs

Let $G$ be torsion-free and hyperbolic relative to a collection $\mathcal{H}$. Let $\operatorname{cd}(G,\mathcal{H})$ be the cohomological dimension of the pair $(G,\mathcal{H})$, and let $\dim$ denote topological dimension. Let $H^q(G,\mathcal{H};\mathbb{Z}G)$ denote the relative cohomology of the pair with coefficients in the group ring, and let $\check{H}^{q-1}(\partial(G,\mathcal{H}))$ denote the corresponding Čech cohomology of the Bowditch boundary. Cohomological dimension and boundary cohomology conjecture. One has

$$\operatorname{cd}(G,\mathcal{H})=\dim \partial(G,\mathcal{H})+1,$$

and, more generally,

$$H^q(G,\mathcal{H};\mathbb{Z}G)=\check{H}^{q-1}(\partial(G,\mathcal{H}))$$

for all integers $q$. The assertion is presented as conjectural in the source; it predicts that the cohomological dimension and relative cohomology of the pair are determined by the topology of its Bowditch boundary.