Hi,
We have this flac file: https://we.tl/t-5ZnMpOASFi
Sorry for the wetransfer link but the file is bigger than 20 MB so GitHub doesn't like it. If it expires before the time you download it, let me know I will send a new one.

We simply load it and set the PublishingDate:

var track = new Track(Path);
track.PublishingDate = new DateTime(2023, 10, 18);
track.Save();

upon reloading and reading the file, the publish date was not saved:

var track = new Track(Path);
Console.WriteLine(track.PublishingDate?.ToShortDateString() ?? ""); //Nothing is printed 

From our testing, this issue only affects flac files and not mp3.