No, the stored energy will change. Only the charge remains constant.
What everyone seems to be overlooking is that there is a force of attraction between the plates. Therefore, mechanical energy (positive or negative) is required to move the plates one way or the other. In this case, the plates are being moved closer together, so energy is removed from the system.
E1 = 0.5 C1 V1^2, E2 = 0.5 C2 V2^2.
So E2/E1 = C2/C1 (V2/V1)^2
In this example we know that C2/C1 = 2 and V2/V1 = 0.5 (since Q is constant)
Therefore E2/E1 = 2 (0.5)^2 = 0.5, ie. half the energy has been removed by the mechanical effort.
QED