I have the vanity with the top handle so I can't store the chains in the bag. My SA said to store it lying down and rest the chains away from the bag. So that's exactly what I did except that I store mine in the box too.
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To be extra safe, you can consider wrapping the chains (that are outside the bag) in the felt/tissue that sometimes comes with the packaging. This way, even if it presses against the leather, it won't make a mark because the chains are wrapped up.
Let us know what you eventually decide to do!