It appears that someone has posted this as a MO question: https://mathoverflow.net/questions/392890/which-unfoldings-o...
As of yet (17.05.2021) there is not an accepted answer.
It would be really nice if we could see a reason (and hence generic proof) why this works. Even more so if you could say why (or why not) the n-cube's (n-1) nets can tile R^(n-1).
I'd like to set a notification for when some hero of future-geometry solves this in ten years.
Do all the 2-nets of the 3-cube tile the plane?
They do!
I haven't been able to find a good diagram of it in a few minutes of searching, but there is a video at: https://etudes.ru/etudes/cubic-parquet/
I can imagine it might be a fun task for a school maths class to try and find all the tilings; you can find the nets of the cube here: https://en.wikipedia.org/wiki/Net_(polyhedron)#/media/File:T...
#/media/File:The_11_cubic_nets.svg){kind=link}
There is a proof on WHUTS that all 261 nets of the 4-cube can, but is a proof by exhaustion (i.e. try every net in an automated manner and see if it is possible), this may be possible to do for the 5-cube nets (9694 possibilities), and 6-cube nets (502110 possibilities), but starts to look difficult in higher dimensions.