Perfect-path density conjecture in the hypercube

Let $Q_d$ be the $d$-dimensional hypercube, and let $P_{d+1}$ denote the vertex set of a path in $Q_d$ with $d+1$ vertices whose endpoints are at Hamming distance $d$. Such a configuration is called a perfect path. Let $\pi(H,d)$ denote the maximum asymptotic density of vertex-induced copies of $H$ in subgraphs of large hypercubes.

Perfect-path density conjecture. For all $d\geq 3$,

$$\pi(P_{d+1},d)=\frac{d!}{(d+1)^{d-1}}.$$

The case $d=3$ is proved in the paper, yielding $\pi(P_4,3)=3/8$. The source also notes that Baber's flag-algebra upper bound for $\pi(P_5,4)$ is consistent with the conjectured value $24/125$.