Libgober's Hodge–Tate conjecture for rationally elliptic quasi-projective varieties

Let $X$ be a quasi-projective variety whose cohomology carries a mixed Hodge structure. The mixed Hodge numbers are denoted by $h^{k,p,q}$.

Libgober's conjecture. If $X$ is rationally elliptic, then its mixed Hodge structure is of Hodge–Tate type; equivalently, every non-zero mixed Hodge number $h^{k,p,q}$ satisfies

$$p=q.$$

The conjecture predicts that rational ellipticity forces the mixed Hodge structure of a quasi-projective variety to be balanced. Its general status is not resolved in the source.