Solvability conjecture for groups with nilpotentizer of order eight

Let $G$ be a group and let $x\in G$. Suppose that the nilpotentizer $nil_G(x)$ has order $8$ and generates a maximal subgroup of $G$. Solvability conjecture. Then $G$ is solvable. This conjecture proposes that the condition $Z^*(G)=1$ can be omitted from Case (1) of Corollary 12 when $p=2$; its status is not established in the supplied text.