Single-degree concentration conjecture for Drinfeld stratification cohomology

Let $r$ divide $n'$ and let $\theta \colon \mathbb T_h(\mathbb F_q) \cong \mathbb W_h^\times(\mathbb F_{q^n}) \to \overline{\mathbb Q}_\ell^\times$ be a character with trivial $\operatorname{Gal}(\mathbb F_{q^n}/\mathbb F_{q^{n_0r}})$-stabilizer. Single-degree concentration conjecture. There exists an integer $i_{\theta,r}$ such that

$$H_c^i(X_h \cap \mathbb L_h^{(r)}\mathbb G_h^1, \overline{\mathbb Q}_\ell)[\theta] \neq 0 \qquad \Longleftrightarrow \qquad i=i_{\theta,r}.$$

This would strengthen the known irreducibility of the alternating sum of these cohomology groups by asserting concentration in one degree. The source gives no resolution or further evidence for the conjecture.