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That's what I'd do, although the tack cloth mentioned above sounds promising.

EDIT: If you choose to sand nicked areas, don't go crazy with it. Do you know what kind of leather it is?

I'm not a strop restoration expert, but here's what I'd do:

Wipe your strop with a *slightly* damp washcloth or shop rag. (A clean one, of course.) Repeat until you don't see any green. Let dry,...