The answer is: yes, if so configured.  When you quit BookMacster, or close a .bmco document window, BookMacster's  agent (BkmxAgent) will continue to sync in the background if you have switched the Syncing button in the toolbar to Ready, indicated by a yellow dot.

         

Synkmark and Smarky users get a warning if the app is not so configured upon quitting, because, unlike BookMacster, syncing/sorting is their only use case.

For all three apps, subsequentlly, provided that you have not disabled it in System Settings / Preferences, macOS will launch BkmxAgent when you log in to your Mac.  So there is no need to run our app.  For syncing usage we recommend that you launch our app only when you want to do bookmarks housecleaning, as explained in this post.

If syncing is not working as expected, please check out our other Top Reasons for Syncing Trouble.