So out of curiosity. What is the burn in phenomenon actually from. I can somewhat understand speaker membranes stretching a little from factory fresh. And possibly amplifier electrolytic caps leakage settling. But what actually led to it being a thing.
Caps should form properly in minutes, and speaker materials ought not to require any signal input to reach equilibrium - it's generally a result of the material "relaxing" after having been stretched or formed during manufacture. Tubes are constantly degrading, so maybe there's a change in operating bias or characteristics after a time which is perceived as "burn-in." This generally only happens in poorly-designed amps, though, because the bias should be stable almost indefinitely until the tube starts to fail.
E.g.: I own a pair of Magneplanars, which are among the few speakers that actually do require some "settling" time - the phenomenon is due to the Mylar diaphragm being mechanically stretched at several thousand psi during assembly and therefore the tension is slightly uneven across the edges. It's a phenomenon called "creep" in the Mylar specification.
Whether you play music through them or not, they will even out after a few weeks and the resonant frequency will drop a bit. If you don't damage or abuse them, they will not change further.
My feeling is that when you hook up a system which is unfamiliar (sometimes poorly matched to the room acoustics, or maybe you just moved the speakers or had it turned off long enough that you've lost any memory of the sonic characteristics) it takes a finite amount of time for you to accustom yourself to the change in sound. It has the same effect as any putative "burn-in" but is purely psychological. There's also expectation bias: if you plug in a new doodad which you're sure is going to change the sound, you will eventually convince yourself that it has improved the sound after a period of time.