Erdős Problem #410 — Growth of iterated divisor sums

Let $\sigma_1(n)=\sigma(n)$, the sum of divisors function, and $\sigma_k(n)=\sigma(\sigma_{k-1}(n))$. Is it true that for all $n\geq 2$ $\lim_{k\to \infty} \sigma_k(n)^{1/k}=\infty?$