It's an old-new question (I found only one similar question with unsatisfactory (for me) answer: Where did Schrödinger solve the radiating problem of Bohr's model?)
It's strange for me how all books simply pass by such an important question, and mentioning strange and mathematically unsupported reasons such as:
orbits are stationary (while as I know this is just idealization, there is no stationary orbits in reality even for Hydrogen)
electrons are actually not localized due to uncertainty principle, thus they have no acceleration (while obviously in a non-spherically symmetric orbits a kind of "charge acceleration distribution" always exist)
vacuum fluctuations play a major role (according to QED).
I'm not interested in how Bohr or Schroedinger explained it, I want to see a rigorous proof with QM, QED or maybe even the standard model as whole. I would like to see how this question was closed.