I had this issue just after moving into a new rental. It seemed to match most of the 'rotten egg and sulfur' smells that are repeated on this thread. I figured out my issue, and may match many of yours as well.
The master bedroom is where they usually install the bulk of the alarm systems, sometimes in a master closet (most of the time it seems). The alarm's backup battery (the little 4 inch square box - black, grey or tan) that it uses can go bad after being constantly charged over many years and it can rupture slightly, emitting a small amount of (you guessed it) sulfuric acid, which really stinks! I made the mistake of sniffing it a bit too close, and that stuff literally takes your breath away. It was bad.
I took it to the backyard where I will dispose of it properly later. It's a hazardous substance, but I seriously doubt it would cause a health concern if it just produces an annoying stench in your closet.
It worked! The stench is gone, gone, gone!
Source: Ol factory experience and old fashioned deductive reasoning.
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