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Leather regardless of what it’s made into needs hydration, all my leather straps get lightly washed when I wash my watch at the end of the week, you don’t want to soak the strap just scrub lightly with a soft bristle brush and some very mild soap, I use Wristclean, anyway after lightly scrubbing and then doing a quick rinse I pat the strap dry then apply a neutral polish like Venetian shoe cream and what ever product you choose to use you don’t need a lot
I never gave it much thought until you asked the question. Now I think it would make sense to condition the leather like I do my shoes....therefore, I've just finished giving my watchbands a coat of Saphir Renovateur.
I use a tiny bit of shoe cream and carefully brush it in, or rub it in with a soft cloth, maybe monthly.
Leather is leather. If shoe cream is good for our shoes, I would think it's fine for watch straps, belts, wallets and purses. It should be stressed, however, to use only small amounts and to brush or rub in in thoroughly, to avoid staining our clothes or skin. Less is more.