A spectral sequence for Dehn fillings

We study how the cohomology of a type $F_\infty$ relatively hyperbolic group pair $(G,\mathcal{P})$ changes under Dehn fillings (i.e. quotients pairs). For sufficiently long where quotient $(\bar{G},\bar{\mathcal{P}})$ is $F_\infty$, we show that there spectral sequence relating groups $H^i(G,\mathcal{P};\mathbb{Z} G)$ and $H^i\left(\bar{G},\bar{\mathcal{P}};\mathbb{Z}\bar{G}\right)$. As consequence, essential cohomological dimension does not increase these fillings.