Cyclic groups minimize the number of connected components of power maps

Let $G$ be a finite group of order $n$, let $C_n$ denote the cyclic group of order $n$, and let $N(a,G)$ be the number of connected components of the functional graph $G(a,G)$ associated with the power map $x\mapsto x^a$. Cyclic-group minimization conjecture. For every $a$, one has

$$N(a,G)\geq N(a,C_n).$$

The conjecture asserts that cyclic groups have the fewest connected components among groups of a given order. The source gives no further evidence for its resolution.