Can someone show me how to prove that $$\liminf_{n \to \infty} \frac{\sigma_{k}(n)}{n} < \infty$$ for every natural number $k$? Or is this problem open? Here, $\sigma_{k}(n)=\sigma(\sigma(\sigma(\dots n)))$ is the $k$-th iterate of the sum of divisors function.

Note: I think for $k=2$ this had been proved by Makowski and Schinzel and the limit equals $1$.

Thank you for any help.