Looking at the diagram of the Lock Ness Monster, it reminds me of a half line, which is infinite in one direction but has a finite starting point. This made me initially think that "1-ended" meant something similar. However, the paper describes the Euclidean plane as 1-ended, suggesting the meaning is quite different.

Although I don't understand it, it appears that the proof answers the original question. However, I wondered if you happened to have an intuitive way of thinking about what "1-ended" means in this context?

The half line is indeed one ended. To be quite honest, the topological definition of ends is a little mystical to me. I generally deal with the ends of hyperbolic spaces, which I am aware are different in some contexts. I'll try to give a mostly intuitive understanding of that, which does apply in this context.

For groups I think of ends as connected components of the ideal boundary. Start by considering all the continuous paths off to infinity. We define a metric on these paths based on how long they stay together before diverging (this can be made formal). This metric gives us the boundary space with points in the space being families of curves that are asymptotically equivalent.

So for the plane this looks like a circle, and for the half line this is a single point. But they are both one ended since those are both connected. The ordinary line however, is two-ended since you either go left or right and there's nothing in between.

I hope that's helpful.

That is helpful - thank you. The connectedness of the end rather than its location gives an intuition for what the Loch Ness monster and the mucube have in common that I couldn't see before.