The generalized Fibonacci cube perfect-code conjecture

Let $n\geq 3$ and $s\geq 1$. A perfect code in $\Gamma_n(1^s)$ is a set $C$ whose vertices have pairwise distance at least three and that dominates every vertex of $\Gamma_n(1^s)$. The hypercube $Q_n$ is the graph on binary strings of length $n$ in which two strings are adjacent when they differ in exactly one coordinate.

Generalized Fibonacci cube perfect-code conjecture. If $C$ is a perfect code in $\Gamma_n(1^s)$, then

$$n=2^p-1$$

for some integer $p$, and furthermore $C$ is a perfect code in $Q_n$.

The preceding results establish existence of perfect codes in $\Gamma_n(1^s)$ when $n=2^p-1$ for sufficiently large $s$, but the minimum such $s$ is not always determined. This conjecture proposes that the existence of any perfect code forces the Hamming-code length condition and that every such code is also perfect in the hypercube.