The element-order divisibility matching conjecture for finite groups

Let $G$ be a finite group of order $n$, let $C_n$ be the cyclic group of order $n$, and write $o(x)$ for the order of an element $x$. Element-order divisibility matching conjecture. There exists a bijection

$$f:G\to C_n$$

such that $o(x)$ divides $o(f(x))$ for every $x\in G$.

This pointwise assertion would imply the corresponding inequalities for sums of inverse powers of element orders. It is presented as a conjecture because the argument in the cited work contains a gap, although the authors report strong evidence for it.