Picard does currently not support writing unsupported tags to MP4 files. Same is true for ASF ( WMA) files.
Some context: Historically Picard did only write the supported tags to all tagging formats, which had more structured tag names (instead of providing just a plain name value list without further specification on the names, e.g. Vorbis tags). This changed at some point for ID3 tags, but not for MP4 / WMA
I think that should be changed, but we have to look at how other software stores custom tags for MP4 to stay compatible. I assume they store it under the “----:com.apple.iTunes:” namespace. You might want to open a ticket on https://tickets.musicbrainz.org