Not sure about CC 123 which is Mute All command in general. In my case with BeatBuddy, it sends SysEx Song Start and Stop commands as described here. HXS responds to these SysEx commands.
Another interesting behavior I found is that even while HXS ignores MIDI clock after receiving SysEX Stop command, if I change tempo at BB via tap tempo, scroll wheel, or MIDI command to BB, then HXS responds to that tempo change and syncs. This tempo change at BB doesn't trigger any MIDI command to HXS but HXS still detects the tempo change. It seems that HXS keeps measuring MIDI clock even when it ignores, and syncs to the clock if tempo change detected.