Thank you very much, tweetpressfr, for reply.
Unfortunately, however, the issue is not resolved. As I can see by the output of the Twitter validator, Twitter has in fact fetched the website and grabbed some data. There is no CDN being used, either.
Again: It is possible to set a custom image URL in a post. However, if you look at the actual card metadata being inserted at the source code of that post, the fallback URL is missing now (as it should) and an image URL is missing completely (as it should not).
Any further ideas on this?