Orders on sets of conformal classes applied to Bartnik's conjecture

In the first part, after showing that the most natural approach to define an order on sets of conformal classes fails, we define a nontrivial order $\leq_2$ on the set of conformal classes of compact Cauchy slabs with fixed past boundary that could help structuring approaches to the Bartnik splitting conjecture via conformal conditions. In the second part we show that if we replace the strong energy condition in Bartnik's splitting conjecture with the null energy condition, then in any dimension greater or equal to $3$ the conclusion of the conjecture would be wrong, more precisely: On a manifold of dimension $\geq 3$, {\em every} globally hyperbolic spatially compact conformal class contains future complete metrics satisfying the null energy condition. In the spatially noncompact case, the same is true in the future of any Cauchy surface.