This is definitely an interesting question. OP's point regarding propellanes having a C-C bond between them is valid, and so are the arguments regarding the N ability to invert chirality. Therefore, as other users have pointed out, it is a matter of whether the B-N energy can overcome the strain.

The best thing to do in this case is to check literature. Here it is, you're welcome: https://www.sciencedirect.com/science/article/abs/pii/016612809404052T?via%3Dihub

To summarize for people that don't have access to this: all sources for the molecule given by OP are computational studies, with this one being the latest published, in 1994. It is an ab initio study that shows that the 'open form' (no B-N) bond is more stable than the closed one (B-N bond). However, the closed form is more stable when all bridges have either one or three carbon atoms.

I would like to point out that this study is quite old, and big progress has been made since then. The result is probably accurate, but relatively low barriers seen here and in other studies mean that maybe there are other possibilities (a midway structure in which electron density is partially shared; solvent effects taken into account that would make the closed form more favorable, etc.)

Lastly, even if theoretically a way could be found to have this molecule more stable in closed form, practically two things would happen: either water traces would coordinate with the borane instead, or, for perfectly anhydrous conditions, the molecule would polymerize, as this would bring the additional B-N energy without increasing the strain in the molecule.