Comparison conjecture for loop Deligne–Lusztig and Drinfeld 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 $\operatorname{Gal}(L/k)$-stabilizer. Let $\chi=\theta|_{\mathbb W_h^1(\mathbb F_{q^n})}$, and assume that the stabilizer of $\chi$ in $\operatorname{Gal}(L/k)$ is the unique subgroup of index $n_0r$. Comparison conjecture. There is an isomorphism of virtual $\mathbb G_h(\mathbb F_q)$-representations

$$H_c^*(X_h, \overline{\mathbb Q}_\ell)[\theta] \cong H_c^*(X_h^{(r)}, \overline{\mathbb Q}_\ell)[\theta].$$

This conjecture seeks to compare the cohomology of the loop Deligne–Lusztig variety $X_h$ with that of its Drinfeld stratification. The paper presents supporting evidence, but the conjecture is not stated as resolved.