I never tried the mini strap. I got a Taylor Suede Guitar Strap for my 814 and loved it. When I got my mini, I ordered one of those for it as well and it works great.
I'm sure the mini strap works great with the mini based on my earned trust of the company ... but I can say from experience that the suede straps definitely do.
I've actually used string for a strap in my younger poorer years
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Enjoy!