The electrode is a closed system, so you can add charge, and it will remain within this closed system.
The central region is a open system, so if there is too much charge in the system, it will run out in the infinite reservoirs,
if there is too little charge, charge will run in from the infinite reservoirs.
So the answer is no. You can't charge the central region as it does not make sense. Add an extra electron would give you the exact same
result as subtracting three electrons, as it would converge to a solution where it has exactly the correct amount of electrons.
This is the entire idea behind NEGF.