The Frobenius bijectivity conjecture for varieties over number fields

Let $X$ be a smooth, geometrically connected, $n$-dimensional projective variety over ${\mathbf Q}$. For all sufficiently large primes $p$, let $X_p$ be a smooth reduction of $X$ over ${\mathbf F}_p$. Frobenius bijectivity conjecture. There are infinitely many primes $p$ such that the endomorphism induced by Frobenius on

$$H^n(X_p,\mathcal{O}_{X_p})$$

is bijective. More generally, the same assertion is proposed for $X$ defined over an arbitrary number field. The source states that this remains open even when $X$ is a curve of genus at least $3$.