On more than one occasion the cork has ripped itself free of the bottle after the cage was removed and then set aside for a couple minutes. Once was in the fridge.
Good to know about the caps. But, does a regular wing-capper work?
You need a 29 mm crownhead, like this one, instead of the (European?) standard 26 mm: https://www.brouwland.com/en/our-products/brewing/beer-bottles-crown-caps/crown-cappers/accessories/d/crownhead-inner-thread-29-mm-for-emily-fiw#.VRKwWxdPZuZ, and, of course, 29 mm caps.
This works on champagne-style bottles but not the Belgian bottles with large mouths like what Jim posted. Those bottles cannot be capped. At least I have never seen a cap or mechanism to apply a cap to those bottles. With my champagne bottles I am content to slap on a 29mm cap without a cork. It might be less classy but the 29mm caps tend to have a thick oxygen barrier and I am comfortable without adding a cork (at least for now).
The red baron wing capper, as I understand it, is the only wing capper capable of capping 29mm caps but you need to buy a separate bell for the larger cap. Many counter or floor corkers have 29mm crownheads available as well.