Just had a good 'google around', can't find anything official but looked at about 10 Q&A sites and everyone says there is no problem. I would think your biggest problem will be the handling of the suitcase so pack it well.
Yes - we've regularly transported bottles of still wine that way, without problems (so far). I'm pleased to hear that the increased pressure difference doesn't cause the fizzy ones to push the cork out, though.
The hold is pressurised so it`s no different to transporting the bottles of champagne that are served in the cabin. It`s knocks that are the main thing to avoid, hence wrapping well.
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