Let $X$ be a $d$-dimensional connected smooth variety over $\overline{\mathbb{F}}_q$.

It is well known that if $X$ is isomorphic to an affine space, then all the $\ell$-adic compactly supported cohomology groups $H^i_c(X,\overline{\mathbb{Q}}_{\ell})$ vanish except $i=2d$. On the other hand, Borel proved the converse when $X$ is a homogeneous space of some connected affine algebraic group (see Theorem 1.4 here) in 1985.

I am wondering if it is known that the converse is true without the assumption on homogeneous, or is there any counterexample?