A few answers on here seem to suggest to something like the following:
In the Many Worlds Interpretation, the amplitude of a branch exponentially decreases over time and so the energy of that branch also decreases exponentially. This seems to suggest the branch amplitude and energy conserved within that branch have a very simple linear relationship.
However, do the amplitudes not just give probabilities for transitions, some of which can be interpreted as measurements of energy? To say that we know the energy has decreased in direct proportion to the branch amplitude is a simplification?