The cohomological dimension formula for Bowditch boundaries

Let $(G,\mathcal{P})$ be a relatively hyperbolic pair of type $F$, with Bowditch boundary $\partial(G,\mathcal{P})$. Let $H^n(G,\mathcal{P};\mathbb{Z}G)$ denote relative group cohomology with coefficients in the group ring.

Cohomological dimension formula.

$$\dim(\partial(G,\mathcal{P}))=\max\{n\mid H^n(G,\mathcal{P};\mathbb{Z}G)\neq 0\}-1.$$

The formula relates the topological dimension of the Bowditch boundary to the highest degree in which relative cohomology with group-ring coefficients is nonzero. The surrounding argument indicates that it follows under the assumptions of the cited theorem, but the supplied text does not establish whether this statement is presented as a conjecture or has been resolved.