15.93 (2002)

Let $G$ be a pro-$p$ group, $p > 2$, and $\varphi$ an automorphism of $G$ of order 2. Suppose that the centralizer $C_G(\varphi)$ is abelian. Is it true that $G$ satisfies a pro-$p$ identity?

An affirmative answer would generalize a result of A. N. Zubkov (Siberian Math. J., 28, no. 5 (1987), 742–747) saying that non-abelian free pro-$p$ groups cannot be represented by $2 \times 2$ matrices.