I wouldn't store ANY bag with its insert inside no matter how soft that insert is, unless I were actively looking to alter its shape.I am a fan of the Whitefox inserts. Soft, not too structured, and wonderfully made. Here are some pictures of my E29 with the insert. The only thing I’ve learned is that clemence leather wants to collapse in. So any insert does raise the risk of creating creases on your leather. So I’ve opted to store my E29 without any insert.
Pic 1 - insert which beautifully follows the natural curve of the base.
Pic 2 - bag lying flat
Pic 3 - bag with insert lying flat. Note the bulging at the top. I should add when the bag is carried vertically, this bulge is less noticeable, but still present.
In summary, whitefox is great, but be aware that Clemence leather can be unforgiving with inserts.
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I do have one Clemence Evelyne PM stored with an insert inside, but it is hanging to prevent the above creasing. The insert is helping to restore a touch of structure to a bag that's had a lot of life. Generally though, storing with inserts will result in marks - even in less soft leathers. Some bag structures *might* be immune to this, but I would not test it.