Let $G$ be a finite 2-group. Let $x$ be a non-central element of $G$ such that
 $C_G(x)\leq cl(x)\cup Z(G)$ where $cl(x)$ denote the conjugacy class of $x$ in $G$. 
Is it true that $|C_G(x):Z(G)|=2$?

**p.s.**<br/>
I know that  if $|G|\leq 2^8$ then the equality holds.