Pippenger–Golumbic's inducibility conjecture for cycles

Let $C_k$ denote the cycle graph on $k$ vertices, and let $\mathcal{I}(C_k)$ be its inducibility, the limiting maximum proportion of induced copies of $C_k$ in graphs with an increasing number of vertices. For $k\geq 5$, Pippenger–Golumbic's inducibility conjecture.

$$\mathcal{I}(C_k)=\dfrac{k!}{k^{k}-k}.$$

The conjecture gives the exact inducibility of every cycle of length at least five. The source states that Pippenger and Golumbic conjectured this lower bound and showed that it is tight for this family of graphs.