Is the longest Hamiltonian path through the $2^d$ unit hypercube vertices known,
where path length is measured by Euclidean distance in $\mathbb{R}^d$?
The unit hypercube spans from $(0,0,\ldots,0)$ to $(1,1,\ldots,1)$.

For example, for the cube in $\mathbb{R}^3$, I believe the longest path
has length
$3\sqrt{2}+4\sqrt{3} \approx 11.17$,
avoiding all edges of length $1$, and using all $3$ of the long diagonals:
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[![LongestCubePath][1]][1]
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<sup>
Path: $(1,7,2,8,3,5,4,6)$.
</sup>
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This likely has been studied, in which case pointers would be appreciated. 
If exact values are not known, bounds would be useful.



  [1]: https://i.sstatic.net/SFxnN.jpg