It is known that $\mathbb{R}^3$ can be partitioned into disjoint circles, or into disjoint unit circles, or into congruent copies of a real-analytic curve (Is it possible to partition $\mathbb R^3$ into unit circles?). But it has recently been proved that the same is not true of Möbius strips:
Frolkina, Olga D. "Pairwise disjoint Moebius bands in space." Journal of Knot Theory and Its Ramifications 27, no. 09 (2018): 1842005. Journal link.
Melikhov, Sergey A. "A note on O. Frolkina's paper 'Pairwise disjoint Moebius bands in space'." arXiv:1810.04089 (2018).
(The first paper proves it for tame subsets, the second removes the tame restriction.) My question is:
Q. Does this new result imply that $\mathbb{R}^4$, or $\mathbb{R}^n$, $n \ge 5$, cannot be partitioned into Klein bottles, congruent or otherwise?
Image from Sensei's Library.
