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
$$frac{1}{n!} sum_{alpha in frak{S}_n} left( sum_{text{irreps} chi} chi(alpha)^2 right)^2.$$

$$sum_{muvdash n}sum_{lambdavdash n}(chi_{mu}^{lambda})^2 =frac{1}{n!} sum_{alpha in frak{S}_n} left( sum_{text{irreps} chi} chi(alpha)^2 right)^2.$$