The strap stays on. Slipping straps are one of my biggest peeves, and in my experience, the day is the only Bal style where the straps stay on my shoulder w/o slipping.
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It really depends on your shoulder type. The Day used to stay on my shoulder, and used to be my all-time favorite style.
But as time went, I realized that I'd been getting back issues since the Day kept slipping off if I didn't tilt my body to keep it on.
So for me, no. It does't stay on my shoulder very well. I've found that the flat shoulder straps (like on the RH Besace and 2002 Flat Belted Hobo) stay on much better cause they are flat, and not rounded like the Day's strap.