The primitive-character L-function formula for the Artin–Schreier hypersurface

Let $F$ be a nonarchimedean local field with residue field of cardinality $q$, let $h$ be a positive integer, let $X$ be the Artin–Schreier hypersurface over ${\bf F}_{q^h}$ described in the paper, and let $\mathscr{L}_\psi$ be the rank-one lisse sheaf on ${\mathbf A}^{h-1}$ associated with a character $\psi$ of ${\bf F}_{q^h}$. Suppose that $\psi$ does not factor through $\operatorname{Tr}_{{\bf F}_{q^h}/{\bf F}_{q^d}}$ for any proper divisor $d$ of $h$. Primitive-character L-function conjecture. Then

$$L({\mathbf A}^{h-1},\mathscr{L}_\psi,t)=\left(1+(-1)^hq^{\frac{h(h-1)}{2}}t\right)^{(-1)^hq^{\frac{h(h-1)}{2}}}.$$

This gives the primitive-character factor in the zeta function of the hypersurface and is intended to determine the corresponding compactly supported cohomology and its Frobenius action. The supplied text does not state whether the conjecture has been proved or disproved.