Finite group in which all partial sum values of squares of degrees of irreducible representations divide the order of the group

Definition
A finite group in which all cumulative conjugacy class size statistics values divide the order of the group is a finite group $$G$$ with the following property: for every natural number $$n$$, the (sum of squares of all degrees of irreducible representations of $$G$$ for which the degree divides $$n$$) divides the order of $$G$$.

To evaluate this property for a group, it suffices to know the defining ingredient::degrees of irreducible representations of $$G$$. Note that we always consider the degrees of irreducible representations over a splitting field such as $$\overline{\mathbb{Q}}$$ or $$\mathbb{C}$$.

Examples
In addition to all finite abelian groups, all groups of order $$p^3$$, $$p^4$$, and $$p^5$$ have this property for any prime number $$p$$.

However, there exist counterexamples of order $$p^6$$ for every prime number $$p$$, so the property is not held by every finite nilpotent group or even by every group of prime power order.

The smallest counterexample is symmetric group:S4, which is a group of order 24 where the sum of squares of degrees of irreducible representations dividing 3 is 20.