Current passed through a charged conductor

To pass a current through the conductor, you connect a source of electrons and a place deficit of electrons, and electrons flow from one to the other.

When you do that, the charge on the conductor will spread throughout the circuit. It will leak some.

If the circuit is grounded, then the charge will all escape.

If there is a switch that can prevent or stop the current from flowing, and if the ground is on the conductor side of the switch, then the charge will all escape whether the current ever flows or not.