Yes, that is exaclty what happens.
Of course, it would take a lot of time to discharge a very big countainer, but the fact that in any container the solution loses conductivity as all ions have adhered to the electrodes is what makes galvanic cells (a.k.a. the basis of the electric battery) not work forever.
With a bit of knowledge on redox reactions you can even calculate how much time it takes for the saline water to lose all its conductivity, but perhaps that would be a question more appropiate for the chemistry stack exchange.
