Conjecture on decomposing even-dimensional hypercubes into paths

Let $n$ be an even positive integer, and let $k$ be a positive integer. The $n$-dimensional hypercube $\mathcal{Q}_n$ has vertex set $\{0,1\}^n$ and edges joining vertices at $\ell_1$-distance one. A path of length $k$ is a sequence of $k+1$ distinct vertices with consecutive vertices joined by an edge. Even-dimensional path-decomposition conjecture. If

$$k\mid n2^{n-1}\quad\text{and}\quad k<2^n,$$

then $\mathcal{Q}_n$ can be decomposed into paths of length $k$. This extends the path-decomposition question to the even-dimensional case, where the degree-parity obstruction forcing $k\leq n$ in odd dimensions is absent. The source presents this as a conjecture without resolving it.