Pippinger's inducibility conjecture for cycles

Let $C_k$ denote the cycle on $k$ vertices. For a graph $H$, its induced density is the number of induced copies of $H$ divided by $\binom{|G|}{|H|}$; write $I(H;n)$ for the maximum induced density of $H$ over graphs $G$ on $n$ vertices, and call $\lim_{n\to\infty} I(H;n)$ the inducibility of $H$. Pippinger's conjecture. For $k\ge 5$, the inducibility of the cycle $C_k$ is

$$\frac{k!}{k^k-k}.$$

This conjecture, posed by Pippinger and Golumbic in 1975, concerns the extremal induced density of fixed cycles. Apart from the case resolved in the paper, the problem remains open; see the cited work for results on the inducibility of other graphs.