It seems that with Plex Sync, in its current form, once you start a sync on a client device you must keep nursing the client device (keep it from falling asleep while a sync is being transcoded, etc.), else your sync will not complete.
I propose that Plex use the push notification services from Google, Apple and Microsoft to periodically notify mobile client devices when there is a sync ready and waiting for download. If a download was started, but never completed, it could send a notification again after a few minutes, waking the client, telling it to take another shot, or pick up where it last lost connection.
This would remove the need for client devices to periodically poll for sync status, since the push notification would now signal when to poll. More importantly, it would ensure that clients always awake and sync, without unnecessary battery consumption, provided they have a usable network connection (connected to the applicable push notification service, and able to connect to plex.tv, and PMS) and have items waiting to be synced.
In addition to this proposed use with Plex Sync, enabling push notifications to mobile clients using the platform's native notification service opens up other possibilities outside the scope of this request, like instant server messaging, etc.
(This would, of course, need to be tied in with ConnectivityManager or Reachability events, or similar, so the client could inform PMS when its current network connection supports syncing, else PMS would keep waking the client up when it has no hope of syncing. (on a slow/metered cellular network, etc.))
An overly simplified diagram showing how this might work:

Communication between PMS and push notification service is abstracted via Plex inc's plex.tv to avoid notification service API keys being distributed in the PMS application.