Koszulness of ideals generated by two-dimensional subspaces

Let $F$ be a field and let $\Theta\subset H^1(G_F,\mathbb Z/2)$ be a two-dimensional subspace. Let $I_\Theta$ denote the ideal generated by $\Theta$ in the cohomology algebra $H^*(G_F,\mathbb Z/2)$. Two-dimensional-subspace ideal conjecture. The ideal $I_\Theta$ is a Koszul module over $H^*(G_F,\mathbb Z/2)$. The source gives no resolution status for this statement and refers to an external definition of Koszul modules.