I have found that if I have, for example, three different custom content type Document Sets added to the same library, then, after creating new Document Set and saving it, Details Pane shows additional columns as not filled, but that columns belong to other Document Set content type in the same library.
If I go back to the document library and reopen the same newly created Document Set again, then all columns are displayed correctly.
The same wrong columns are displayed when link to document set is opened from Flow dynamic content called "Link To Item".
I have noticed that CORRECT columns are displayed when URL in browser has "viewid=".
And INCORRECT columns are showed when URL doesn't have "viewid=" - just after saving the document set, or reaching it from MS Teams Dynamic content link.
I have already reported the issue, MS has opened the ticket, but still waiting for an answer.
Also it looks like when I add first custom Document Set content type to Document Library, everything is correct. But when I add second one or third one, and try to create new Set, the additional fields from first content type is show as not filled out.