Reselect your vertical path, focus on the Appearance panel, make sure that the existing stroke is selected and duplicate it using the Duplicate Selected Item button (pointed by the blue circle in the following image). Select the new stroke, replace the existing linear gradient with the one shown in the following image, set its Weight at 1px and go to Effect > Distort & Transform > Transform. Drag the Move-Horizontal slider at 0.5px and click OK.