Conjecture on special-cycle generation of cuspidal cohomology

Let $Y$ be the arithmetic manifold, let $E(\lambda)$ be the coefficient system, and let $H^{n\times q}_{\mathrm{cusp}}(Y,E(\lambda))$ denote the indicated cuspidal cohomology subspace. Special-cycle generation conjecture. Theorem remains valid as long as $p>2n$ and $m-1>3n$. If $p=2n$ or $m-1\leq 3n$, the space

$$H^{n\times q}_{\mathrm{cusp}}(Y,E(\lambda))$$

is not spanned by projections of classes of special cycles.

The statement proposes the sharp range for generation by special cycles and predicts failure outside that range; the supplied text gives evidence but no resolution.