The cohomology conjecture for powerful 4-star-central 2-groups

Let $G$ be a powerful $4^*$-central $2$-group with the $\Omega$EP property, and let $d=\operatorname{rk}(\Omega_1(G))$. Write $\Lambda$ for the exterior algebra. The cohomology conjecture for powerful $4^*$-central $2$-groups. Then

$$H^*(G;{\mathbb F}_2)\cong \Lambda(y_1,\ldots,y_d)\otimes {\mathbb F}_2[x_1,\ldots,x_d].$$

Here the generators $y_i$ have degree one and the generators $x_i$ have degree two. This is proposed as the cohomological structure for powerful $4^*$-central $2$-groups with the $\Omega$EP; the source gives no resolution.