I would like to know under what conditions there exisits a (possibly open) knight's tour on a generic (hyper)cubic lattice $\{\{0,1,\ldots,n-1\} \times \{0,1,\ldots,n-1\} \times \cdots \times \{0,1,\ldots,n-1\}\} \subseteq \mathbb{Z}^k$, where $n \in \mathbb{N}-\{0,1,2,3\}$ and $k \in \mathbb{N}-\{0,1\}$ (about the definition of closed/open knight's tour, see An efficient algorithm for the Knight's tour problem).
As far as I know, in 2012, this problem has been solved by Erde, J., Golénia, B., and Golénia, S. (see "The closed knight tour problem in higher dimensions", The electronic journal of combinatorics, no. 19(4), #P9) for any $k$, but under the additional constraint of considering only closed knight's tours (instead of an open or closed knight's tours).
Has the open knight's tour problem been solved for any $k \geq 3$ (I am not interested in a constructive solution, my concern is only about the existence of such a self-crossing covering path with link-length $n^k-1$)?
Any reference about the $k$-dimensional version of the well-known knight's tour problem is welcome.