No. Because of a few different things: 1. There's no rule saying that the supply box transfers from the pilot to the TAG. 2. The supply box rules require that supply box token is always kept on the table, and when the pilot mounts the TAG, you remove it from the table. 3. The TAG isn't a specialist troop. The pilot is. In other words, when the specialist trooper pilot mounts the TAG, you leave the supply box on the table and the TAG isn't carrying it. Or the specialist trooper pilot can't mount the TAG.
A TAG and its Pilot is the same unit with different profiles, just like a Motorcycle trooper. I think the easier answer here is thus; No, because the Pilot is a different model from the TAG and the model is the one carrying the box. When entering the vehicle, the box' token is kept on the table, but is not carried by anyone. For a Motorcycle the answer is yes because the trooper's model is replaced with the mounted or dismounted model and as such the Trooper will continue carrying the box.
I'm with mahtamori in this. If states affecting the pilot pass to the TAG and viceversa. Shouldn't we consider the tokens follow the exact same procedure?