For anyone else who winds up seeing this, it’s not .NET that needs Wine. You can use .NET to write Linux programs.
The problem is that the techniques for making an oddly-shaped, borderless window (the goose) that can move other windows are very specific to each desktop, and the author only wrote the code to do it on Windows and macOS. (Wine is a translation layer for allowing Windows programs to talk to Linux as if they’re running on Windows.)