The celebrated positive mass theorem, first proven by Schoen and Yau and then by Witten, states that the ADM energy-momentum vector of an asymptotically flat initial data set with the dominant energy condition must be either future timelike or null. This led to a natural conjecture characterizing initial data sets with null ADM energy-momentum, known as the equality case. It turns out that this conjecture is interconnected with the problem of mass minimization in the context of the Bartnik quasi-local mass, known as the stationary conjecture. I will describe a variational approach to advance both conjectures, as well as counterexamples in higher dimensions without the optimal decay rate for asymptotically flatness. These talks are based on a series of joint works with Dan Lee.