Bollobás–Meir conjecture for the power-weighted Euclidean traveling-salesman problem

For a finite set of points $X \subseteq [0,1]^k$, let $S_k(H)=\sum_{e\in H}|e|^k$ for a Hamiltonian cycle $H$ on $X$, and let

$$S_k^{\mathrm{HC}}(X):=\min_H S_k(H),\qquad S_k^{\mathrm{HC}}:=\sup\{S_k^{\mathrm{HC}}(X):X\subseteq[0,1]^k,\ X\text{ finite}\}.$$

Bollobás–Meir conjecture. For any finite set of points $X\subseteq[0,1]^k$, there exists a Hamiltonian cycle $H$ on $X$ with $S_k(H)\leq 2\cdot k^{k/2}$ if $k\neq3$, and $S_3(H)\leq2^{7/2}$ if $k=3$. Equivalently,

$$S_k^{\mathrm{HC}}=\begin{cases}2\cdot k^{k/2}&\text{for }k\neq3,\\2^{7/2}&\text{for }k=3. \end{cases}$$

The original Bollobás–Meir conjecture asserted the first value for every $k$, but it was disproved in dimension $3$ by Balogh, Clemen and Dumitrescu. The displayed statement is their adjusted conjecture, which remains open for all $k>2$; the cases $k=1,2$ are known.