Conjecture on extremal Hamiltonian paths in the unit cube

Let $X$ be a finite point set in the unit cube $[0,1]^k$. For a graph $G$ on vertex set $X$, define its scaled cost by

$$s_k(G)=\left(\sum_{e\in G}|e|^k\right)^{1/k}.$$

Let $s_k^{\texttt{HP}}(X)$ be the minimum scaled cost over Hamiltonian paths on $X$, and set

$$s_k^{\texttt{HP}}=\sup_{n\geq 2}\sup_{X\subseteq[0,1]^k,\,|X|=n}s_k^{\texttt{HP}}(X).$$

Hamiltonian-path conjecture. The equalities

$$s_k^{\texttt{HP}}=\begin{cases}\sqrt{3},&k=2,\left(2^{k-1}-1\right)^{1/k}\sqrt{2},&k=3,4,5,6,\sqrt{k},&k\geq 7\end{cases}$$

hold.