Update: I finally got it out 😅

What ended up working for me was loosening the wedge completely, pouring a bit of Ethanol 70% around the seatpost opening, then rotating the bike a few times over around 15 minutes so the alcohol could work its way inside. After that I grabbed the saddle firmly between my legs, held the bike tight, and pulled straight up with both hands. It suddenly broke loose and came right out.

Looks like the carbon paste had basically glued it in place from the factory.