Conservation of energy requires a well-defined dynamics that has the requisite properties (basically that the Lagrangian or Hamiltonian is independent of time). Generally speaking that is true for unitary dynamics in QM, but it is not true for collapse because collapse has no well-defined dynamics; it's just declaring by fiat that the wave function changes discontinuously. So one would not expect conservation laws to hold in such a case.
However, it should be noted that, in basic QM (i.e., without adopting any particular interpretation--discussion of how particular QM interpretations deal with collapse belongs in the interpretations subforum), "collapse" is not claimed to be an actual physical process; it's just an update you make in your mathematical model when you know the result of a measurement. Since making a measurement on a quantum system requires the system to be open, i.e., it has to interact with other systems, one would not expect conservation laws to hold for the measured system alone, since conserved quantities can be exchanged through the interaction. For example, the measured system could gain or lose energy from the interaction that takes place during meaurement, so its energy taken in isolation would not be conserved. Only the energy of the whole larger system, including the measuring device and anything else the measuring device interacted with, would be conserved.
The paper you reference (which, btw, is a preprint and it's not clear whether it has actually been published) does not appear to take such things into account. Its view of "collapse" also appears to be interpretation dependent--it treats it as an actual physical process, which, as above, is not how basic QM independent of any intepretation treats it.