As I sort of assumed, there is no real consensus on this. That said, I love Hassles' response above.
I practice a lot in the evening and I imagine pads and the interior of the horn takes hours to dry completely, even if completely exposed to air and properly swabbed.
In my case, the best approach might be to leave it on its stand overnight and put it in the case when I wake up, but that would rarely if ever happen. I'm not compulsive enough.
That said, the horn has done pretty well for itself since it was made in 1955. I imagine it was probably through a lot in the beatnik and hippie eras. Maybe my question is a bit compulsive.