The Prim'–Cayley fixed-point sum conjecture

Let $\operatorname{Prim}'$ and $\operatorname{Cay}$ be the species of primitive objects and Cayley permutations, respectively. For a Cayley permutation $w=w_1\cdots w_n$, let

$$\textsc{fix}(w)=\{i:w_i=i\}$$

be its set of fixed points. The Prim'–Cayley fixed-point sum conjecture. For all $n\geq 1$,

$$\frac{1}{2}|\operatorname{Prim}'[n]|=\frac{1}{n}\sum_{w\in\operatorname{Cay}[n]}\sum_{i=1}^n w_i=\sum_{w\in\operatorname{Cay}[n]}\sum_{i\in\textsc{fix}(w)}i.$$

The conjecture concerns a further relationship between the species $\operatorname{Prim}'$ and $\operatorname{Cay}$ beyond the known species equality $\operatorname{Prim}'=\operatorname{Cay}^2$. The paper reports that no bijective proof of that equality was found; no resolution of this fixed-point sum conjecture is given.