Read the first section of this paper. Conservation of energy absolutely is a problem for objective collapse theories.

The definition of conservation being employed in the paper is this: The probability distribution of the eigenvalues of a conserved quantity must remain constant. If this condition isn't satisfied, it's hard to see why one should consider the quantity conserved.

ETA: I can also give you a non-technical heuristic argument against conservation of energy during collapse. When a particle's position-space wavefunction collapses, its momentum-space wavefunction must spread out in accord with the uncertainty principle. In the aggregate, this corresponds to increase in the average squared momentum, which in turn corresponds to an increase in kinetic energy. So collapse produces an increase in energy out of nowhere.