This question is related to the earlier question which is in the given link: Primary invariants of a finite group

Let $G$ be a finite group and $V$ a complex representation of degree $n$, and let $f_1,f_2,\ldots,f_n \in Sym(V)^G$ be algebraically independent invariants.

Suppose the $f_i$'s satisfy the condition that $\frac{\prod deg(f_i)}{|G|}$ is an integer and also least possible.

Does that mean $f_i$'s are a homogeneous system of parameters?