General conjecture on the optimal covering trails in a \(k\)-dimensional cubic lattice

We introduce a general conjecture involving minimum-link covering trails for any given \(k\)-dimensional grid \(n \times n \times \cdots \times n\), belonging to the cubic lattice \(\mathbb{N}^k\). In detail, if \(n\) is above two, we hypothesize that the minimal link length of any covering trail, for the above-mentioned set of \(n^k\) points in the Euclidean space \(\mathbb{R}^k\), is equal to \(h(n, k) = \frac{n^k − 1}{n − 1} + c \cdot (n − 3)\), where \(c = k − 1\) iff \(h(4, 3) = 23\), \(c = 1\) iff \(h(4, 3) = 22\), or even \(c = 0\) iff \(h(4, 3) = 21\).