I never used this kit to study adiponectin, but I did plenty of western blot for adiponectin some years ago. I remember that freezing the cell lysates (I used to store only at -20 for short periods) did not seem to affect the signal. So my guess is that it should be OK to store the supernatant at -80 until use.
Can you please share the catlog number? We use some differentiation kit, which has a media and suppliment. The media is kept at 4 degree and suppliment at -80.