demonstration using parseval
Your first steps seem OK to me. Even if I'd put absolute values in the time domain as well (for generality), if the first three integrals exist, the fourth is valid. Integration is linear.

The weak point resides in the reverse inference. The equality of energy integrals does not mean that the integrands are related. For one, PDSs are real and positive functions. And there is NO reason why the Fourier transform of a sum of squared functions should be real and positive in general.

To convince your advisor, I'd suggest you to build your question on a clever counter-example.