Disclaimer: my grasp of differential/Riemannian geometry is not quite strong enough to answer your underlying question, but I think I understand all the various pieces enough to answer this question on meta.
First off: What is "Fact or Fiction?" doing in the question title? It comes across as click-baity and adds nothing to the actual question, which appears to be "Can this geodesic-closure construction be embedded in the three-torus?" I am awaiting the day when someone posts "Ten integrals maths teachers don't want you to be able to solve! (You won't believe the $\tan^{-1}$ in number $3$)." but so far it hasn't happened.
Next: this question is probably too short. While keeping things concise is a good idea in general, an old writing rule that applies in maths too is "Say everything you need to say in only the words needed to say it." I think your question would benefit from describing the geodesic closure construction that you're talking about, and exactly (mathematically) how you're putting these things into the cube (is that in ${\mathbb R}^3$ by the way?) since any answer will need to consider exactly that. When I read your question I had to go the linked paper to check the geodesic closure (section 2.3) and I still wasn't very sure how you're putting them in the cube or connecting the points at infinity to the corners of the cube.
Rather than remove what you already have though, I'd keep it as an executive summary of the question, so that potential answerers can assess how difficult your question is.
Next: you may be asking on the wrong site. This might just be better asked on mathoverflow.com.
Finally: you're lacking context and motivation. Even if this is just because it's of intrinsic interest to you, it would be better to state that, but asking if it can be embedded in the three-torus suggests that you have a specific application in mind: so tell us :)
And so that you're not the only one I'm criticising here, the commentor, who appears to be a high-rep-worth individual with plenty of answers in the relevant tag, could certainly have taken one more sentence to explain why they think what you've written isn't good. If they understand the area well enough to know what the problems are, then they understand it well enough to point them out. Maybe they were busy and will provide an update later, but as it stands their comment is unhelpful, and might be taken by some to be rude.
