After a NetSuite shipment is booked in, the tracking number will only pass back to NetSuite if the trigger status is set to "Packed" or "Shipped".
If your trigger status is "Picked", the tracking number will not be passed back. This is because when an order is picked, there are no packages present on the order, which is where the tracking numbers are stored.
If you have any questions regarding this or anything else, please don't hesitate to get in touch with the Shiptheory support team!