A remark on descent for Coxeter groups

Let $\Gamma$ be a finite Coxeter group with reflection representation $R$. We show that a $\Gamma$-equivariant quasicoherent sheaf on $R$ descends to the quotient space $R//\Gamma$ if it descends to the quotient space $R//\langle s_i\rangle$ for every simple reflection $s_i\in \Gamma$.