The good news is that you have tracking.
The buyer is the one that has to close it "by hand" it isn't automatic.
Update the case with the tracking number ASAP. This puts the "action" within the case back on the buyer. I suggest putting in the updates as it makes progress each day, use the "message" function in the case, each time that returns the action to the buyer if they happen to respond within the case.
If it arrives the buyer wouldn't ask eBay to step in. I am not 100% sure if the bots would automagically close against you if the action is on the buyers side on the 16th, I suspect not, but someone else may know for sure.