Assume the Carmichael's Totient Function Conjecture.

Consider the totient chain

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

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

(1) How fast does this chain grow?

(2) Is there a prime $p$ with $p|n_{i+1}$ and $p\nmid n_i$ and $p>q$ for every prime $q$ with $q|n_i$?

What if $3^2$ is replaced by $a^k$ where $a,k+1\geq3$?

I think the growth should be at least doubly exponential.