Koszulness of annihilator and extension-cohomology modules

Let $\ell$ be a prime number and let $F$ be a field containing a primitive $\ell$th root of unity. For $u\in H^1(G_F,\mathbb Z/\ell)$, let

$$\operatorname{Ann}(u)\subset H^*(G_F,\mathbb Z/\ell)$$

be its annihilator, and for a cyclic extension $E/F$ of degree $\ell$, let $H^{\ge 1}(G_E,\mathbb Z/\ell)$ be regarded as a module over $H^*(G_F,\mathbb Z/\ell)$. Koszulness conjecture. For every such $u$, the module $\operatorname{Ann}(u)$ is Koszul over $H^*(G_F,\mathbb Z/\ell)$; for every such cyclic extension $E/F$, the module $H^{\ge 1}(G_E,\mathbb Z/\ell)$ is Koszul over that algebra. The source refers to an external definition of Koszul modules and gives no resolution status for this statement.