Cart disappears at checkout pt2x

So I have searched a bit and see that this is not only happening to me.

I have a multisite, and (so far) one shop installed there.

http://www.shopjust.org/chotika

On my Chrome, the shoppingcart works, and I can go to checkout.
Only issue is that the personal info from the last customer stays in the checkout form.

which is not very sexy...

On all other browsers I have tested, the shoppingcart goes empty when I hit checkout.

I have searched a bit and see that I am far from alone.

Currently only MarketPress is activated...

From what I see, it could be a cookie setting.

But I have not really figured out how to fix that.

I will try to reinstall MarketPress, but I am afraid of losing the items in the shop...

This is kinda a showstopper...