Assume the [Carmichael's Totient Function Conjecture](https://en.wikipedia.org/wiki/Carmichael%27s_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 $a,k+1\geq3$?


It is clear if $2^k$ were used then the chain growth is given by approximately $n_i\approx 2^{2^i}$.