One of the main problems is to study this as a mental experiment or a thought experiment. That is alsways dangerous as if you can reach final agreements.
What is much more important to describe a real experiment in all its details, as much as possible, with all its complications, such that it can be performed in real.
Real experiments for example always involve accelerations, gravity. Also, at least, two objects are involved and space cannot be considered empty.
The importance is, if there exists a now, than two events which happened at the same instant, also happened at the same time.
The 'problem' is that they should also happen at the same clock reading.
This is the issue discussed in the document:
Article_Review_On The Electrodynamics Of Moving Bodies Specific in the Appendix of that document.
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