Well this one got me thinking, in the first few minutes I managed to convince myself that the scales would measure the weight of the box only and then I convinced myself that they would measure both the box and helicopter, then I wasn’t sure Ahhhhhh!!!
However several hours later, I think this explanation answers it for me: -
As TC stated, the downwash will strike the bottom of the box and exert a force on it. However, that is an oversimplification because the air will now dissipate sideways, strike the side of the box deflect upwards hit the top of the box and then get sucked back down through the rotor system. At this point there will be a reduction in pressure on the ‘inside’ of the top of the box and an increase in pressure on the base of the box, hence there will be a greater force exerted by the air on the bottom of the box than on the top of the box. The resultant of these two forces will be a force downward (onto the scales) equal to the weight of the helicopter.
The scales will show the weight of the helicopter and the box.
That’s my theory for what it is worth.
Cheers
TeeS