The nonvanishing conjecture for cuspidal cohomology of $GL(n)$

Let $F$ be a number field, let $G=GL_n$, and let $E$ be a coefficient field. Write $X^+_{00}(\operatorname{Res}_{F/\mathbb{Q}}(T_0)\times E)$ for the set of strongly-pure dominant weights, let $K_f$ be an open compact subgroup of $G(\mathbb{A}_f)$, and let $\mathcal{S}^G_{K_f}$ and $\widetilde{\mathcal{M}}_{{}^\iota\lambda,\mathbb{C}}$ denote the associated locally symmetric space and coefficient system for an embedding $\iota:E\to\mathbb{C}$. Nonvanishing conjecture. Suppose $\lambda\in X^+_{00}(\operatorname{Res}_{F/\mathbb{Q}}(T_0)\times E)$ is a strongly-pure weight. Then for some sufficiently deep open compact subgroup $K_f$ of $G(\mathbb{A}_f)$,

$$H_{\rm cusp}^\bullet(\mathcal{S}^G_{K_f},\widetilde{\mathcal{M}}_{{}^\iota\lambda,\mathbb{C}})\neq 0,$$

for every embedding $\iota:E\to\mathbb{C}$. This is the basic nonvanishing problem for cuspidal cohomology with coefficients in strongly-pure weights for $GL(n)$ over a number field; the source provides no resolution status, so the conjecture remains open.