The Borel–Wallach range vanishing conjecture for localized cohomology

Let $E$ be the quadratic field under consideration, let $K_f$ be the chosen level, let $\mathcal{L}_{\bm{n}}(\mathbb{F})$ be the coefficient system, and let $\mathfrak{m}$ be a non-Eisenstein maximal ideal of the Hecke algebra. Let $b_E$ and $t_E$ denote the endpoints of the Borel–Wallach interval. Borel–Wallach vanishing conjecture. The cohomology groups

$$H^i(Y_E(K_f),\mathcal{L}_{\bm{n}}(\mathbb{F}))_\mathfrak{m}$$

vanish unless $i\in[b_E,t_E]$. This vanishing is a conditional input for the Calegari–Geraghty method; the source gives no proof of the assertion in the stated setting.