I read references 2,3
3 is quite readable.
2 is tougher and I am not through.
The idea that some local properties can't be extended globally in some topologies is not as strange as it might seem at first glance; there are other examples.
I do have doubts about some of the reasoning; but that doesn't fault the reasoning just the presumptions.
I think its possible that the "energy conditions" are not the right analysis tool. Something along the lines of Entropy (Maxwell's daemon ) might be sharper in the mathematical sense.