Conjecture about the prime number function

Question

Define $p_{n,m}$ by $p_{n,1}=p_n$ and $p_{n,m}=p_{p_{n,m-1}}$,
where $p_n$ is the prime number function.

Conjecture:
$n^n<p_{n,n}< (n+1)^{(n+1)}$

I've only tested it for $n=1,\dots,11.$

I need help testing the conjecture for $n>11$ and to prove it. Perhaps the conjecture is a consequence of PNT?

 n         n^n     p_{n,n}  (n+1)^{n+1}
 1            1             2             4
 2            4             5            27
 3           27            31           256
 4          256           277          3125
 5         3125          5381         46656
 6        46656         87803        823543
 7       823543       2269733      16777216
 8     16777216      50728129     387420489
 9    387420489    1559861749   10000000000
10  10000000000   64988430769  285311670611
11 285311670611 2428095424619 8916100448256

Calculate $p_{n,m}$ by m times iteration:
1. $p_{n,1}=p_n$
2. $p_{n,2}=p_{p_{n,1}}$
..
m. $p_{n,m}=p_{p_{n,m-1}}$

Answer 1

Using a lower bound $p_n>n\log n$, one can show that $\log{p_{n,m}}>\log{p_{n,m-1}}+\log\log{p_{n,m-1}}$. By computer, one can check that $\log{p_{26,26}}>89.0>\log27^{27}$.

It looks like always $\log{p_{n,n}}>\log(n+1)^{n+1}$ for $n\ge26$, but I can't prove it...