You're doing everything correctly, but you've just found a bug
I can confirm that I see the same issue on my end, and the "good" news is that it's limited to changes made with Audio Editor.
If instead I make timeline changes (like changing the volume of the clip) or use the stand-alone AudioDirector and then send the edited track back to PD, everything works normally when saving and reopening the project.
In your case, is it possible to just lower the volume on the music track? With the audio track height set fairly high you can get a reasonably fine control by clicking on the thin white horizontal line on the clip on the timeline and dragging it down.
You can also open the Keyframe tool with the playhead at the start of the clip then expand the Volume control. Type in the dB level you want (including a negative sign if it will be less than 0) then move the scrubber to the end of the track and type in the same number to set the starting and ending keyframes to the same level. These changes are correctly saved.
If you truly need the dynamic range compression (or any other Audio Editor edits), when you're finished you'll need to produce just the music track to an audio-only format like WAV. You can do that by temporarily removing all other timeline content and producing the clip, or move the clip to the end of the timeline and use the Range Selection tool then Produce Range.
Either way, replace the original music clip with the newly-produced one to lock in the volume changes until CL patches the issue. You can report it directly to them from this
page, and make sure you include a link to this discussion. I can also file a report if you prefer.