The compositum conjecture for the fourth p-Zassenhaus layer

Let $p=2$, let $F$ be a field of characteristic different from $2$ containing a primitive second root of unity, and let $F(2)$ be its maximal $2$-extension. Let $F_{(4)}$ denote the fixed field of the fourth term of the $2$-Zassenhaus filtration of $G_F(2)$. Let $C_2$ and $C_4$ be cyclic groups of orders $2$ and $4$, let $D_4$ be the dihedral group of order $8$, and let $G_1,G_2$ be the specified finite groups. Then the compositum of all $C_2,C_4,D_4,G_1,G_2$-extensions $K/F$ inside $F(2)$ equals $F_{(4)}$.

Fourth-layer compositum conjecture.

$$F_{(4)}=\text{the compositum of all }C_2,C_4,D_4,G_1,G_2\text{-extensions }K/F\text{ inside }F(2).$$

The assertion generalizes the known description of $F_{(3)}$ as the compositum of the $C_2$, $C_4$, and $D_4$ extensions. The source establishes the inclusion from the indicated compositum into $F_{(4)}$, but the reverse inclusion remains conjectural.