reference request – Total sum of squares of characters of the symmetric group $frak{S}_n$

In my earlier MO post, I proposed the double sum $sum_{muvdash n}sum_{lambdavdash n}chi_{mu}^{lambda}$ regarding characters of the symmetric group $frak{S}_n$. Soon after, I started considering the sum of squares $sum_{muvdash n}sum_{lambdavdash n}(chi_{mu}^{lambda})^2$ hoping to gain a better formula. A further look into older MO posts here and also here shows a Burnside-type Lemma
$$frac{1}{n!} sum_{alpha in frak{S}_n} left( sum_{text{irreps} chi} chi(alpha)^2 right)^2.$$

I would like to ask:

QUESTION. The numerics suggest the below equality. Why is this true?
$$sum_{muvdash n}sum_{lambdavdash n}(chi_{mu}^{lambda})^2
=frac{1}{n!} sum_{alpha in frak{S}_n} left( sum_{text{irreps} chi} chi(alpha)^2 right)^2.$$