Assume the Carmichael's Totient Function Conjecture.

How fast does this chain grow?

$$n_0=(\phi(3^2))\rightarrow n_1=(\phi(\phi^{-1}(n_0))^2)\rightarrow n_2=(\phi(\phi^{-1}(n_1))^2)\rightarrow\dots$$

where $\phi^{-1}(\phi(x))$ is the smallest second integer $y\neq x$ such that $\phi(x)=\phi(y)$.