In this post, for integers $n\geq 1$ we consider the sum of divisors function $\sum_{d\mid n}d$ denoted as $\sigma(n)$ and the divisor-counting function $\sum_{d\mid n}1$ as $\sigma_0(n)$. Then I wondered about the solutions of $$\sigma\left((\sigma_0(n))^4\right)=n.\tag{1}$$

Here is the Wikipedia's article Divisor function dedicated to the sum of the positive divisors of an integer $n\geq 1$, that is $\sigma(n)$ and the arithmetic function $\sigma_0(n)$ (also denoted in the literature as $\tau(n)$) that counts the number of those positive divisors.

I was inspired in [1].

Question. Can you prove or refute that there exist infinitely many solutions $n\geq 1$ of $$\sigma\left((\sigma_0(n))^4\right)=n\,?$$ Many thanks.

Computational evidence. Using a Pari/GP program I know that the first few solutions of previous equation $\sigma\left(\sigma_0(n)^4\right)=n,$ are $$1 ,31 ,121 ,511, 3751$$ and $61831.$ Thus I've no enough computational evidence showing that maybe should be infinitely many solutions of $(1)$. Finally seems that previous sequence isn't in the OEIS.

References:

[1] Problem 3565, Crux Mathematicorum, Volume 37, Number 6 (2011).