Boyarchenko's maximality conjecture for the variety $X_h$

Let $X_h$ be a finite-type variety over a finite field with geometric Frobenius $\Fr_{q^n}$ acting on its compactly supported cohomology. A variety is maximal in the sense of Boyarchenko--Weinstein when its cohomology has the prescribed parity vanishing and Frobenius eigenvalues. Boyarchenko's maximality conjecture. $X_h$ is a maximal variety in the sense of Boyarchenko--Weinstein. Equivalently,

$$H_c^i(X_h, \overline{\mathbb{Q}}_\ell)=0$$

unless $i$ or $n$ is even, and the geometric Frobenius $\Fr_{q^n}$ acts on $H_c^i(X_h, \overline{\mathbb{Q}}_\ell)$ by the scalar $(-1)^i q^{ni/2}$. This conjecture gives precise control of the cohomological parity and Frobenius weights needed to analyze the cohomology groups attached to semi-infinite Deligne--Lusztig varieties; its resolution is not specified in the source.