Perfect-cycle density conjecture in the hypercube

Let $Q_d$ be the $d$-dimensional hypercube, and let $C_{2d}$ denote the vertex set of a $2d$-cycle in $Q_d$ whose $d$ opposite pairs of vertices are at Hamming distance $d$. Such a configuration is called a perfect $2d$-cycle. Let $\pi(H,d)$ denote the maximum asymptotic density of vertex-induced copies of $H$ in subgraphs of large hypercubes.

Perfect-cycle density conjecture. For all $d\geq 4$,

$$\pi(C_{2d},d)=\frac{d!}{d^d}.$$

The cases $d=3$ and $d=4$ are addressed by the paper's results, with the displayed theorem giving $\pi(C_8,4)=3/32$; the conjecture concerns the general range $d\geq4$.