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9/16! How big are your hands, man! :D
Careful what you use on the case. Oil will probably darken it significantly. It'd be a shame to change it. I can't advise on what would be best.
Enjoy them.

That case is a work of art! Hope the leather's still nice & supple.
What is that, 3/8? I have a tiny Helje looks about that size. Nothing stands in it's way!