The primitive-character cohomology conjecture for the Lubin–Tate hypersurface

Let $H^B$ be the finite nilpotent group acting on the hypersurface $X$, let $V_\psi$ be the unique representation of $H^B$ lying over a character $\psi$ of its center, and let

$$\mathcal{H}=H_c^*(X\otimes\overline{\mathbf F}_q,\overline{\mathbf Q}_\ell)$$

be regarded as a virtual module for $\operatorname{Gal}(\overline{\mathbf F}_q/{\mathbf F}_{q^h})\times H^B$. Assume that $\psi$ does not factor through $\operatorname{Tr}_{{\mathbf F}_{q^h}/{\mathbf F}_{q^d}}$ for any proper divisor $d$ of $h$, and define

$$\mathcal{H}_\psi=\operatorname{Hom}_{H^B}(V_\psi,\mathcal{H}).$$

Primitive-character cohomology conjecture. As a virtual module for $\operatorname{Gal}(\overline{\mathbf F}_q/{\mathbf F}_{q^h})$, one has $\dim\mathcal{H}_\psi=(-1)^{h-1}$, and the eigenvalue of $\operatorname{Frob}_{q^h}$ on $\mathcal{H}_\psi$ is $q^{h(h-1)/2}$. This is an alternate representation-theoretic formulation of the paper’s main conjecture, describing the multiplicity and Frobenius eigenvalue of the primitive representation in the compactly supported cohomology. The supplied text does not state whether the conjecture has been proved or disproved.