Generalized middle levels conjecture for hypercube level subgraphs

Let $Q_n$ be the $n$-dimensional hypercube, and let $Q_{n,[a,b]}$ denote the subgraph induced by the vertices whose levels lie in $[a,b]$. For integers $k\geq 1$ and $c\in\{0,1,\ldots,k\}$, consider $Q_{2k+1,[k-c,k+1+c]}$. Generalized middle levels conjecture. For any $k\geq 1$ and $c\in\{0,1,\ldots,k\}$, the graph $Q_{2k+1,[k-c,k+1+c]}$ has a Hamilton cycle. The case $c=0$ is the middle levels conjecture, which was announced as solved recently; the generalized cases were presented as remaining open and are used to reduce the other open instances of the enumeration problem.