Phased array beam-forming requires a minimum distance between transmitters in order to both form the wave and to reduce the mutual coupling between antennas. (Mutual coupling is a near-field phenomena that can significantly degrade antenna performance: for example, mutual coupling = -3dB means 50% of the transmitted power form one antenna is absorbed back into the circuit of the adjacent antenna).
Most antenna phased array systems assume a minimum separation distance of half-wavelength between antennas in order to achieve mutual coupling below -15 dB. At 2.4 GHz, the wavelength is roughly 12cm, so the minimum separation distance in free space is 6cm. Assuming they use FR4 as the PCB material to support the antennas, the physical separation can be reduced to roughly 3cm between antennas (Rogers is another material with dielectric constant of 10, but very expensive). The current size of their transmitter is 184cm tall, so it is conceivable that if the width is at least 6cm, it is possible to contain 200 transmitters. (However this is basically a vertical phased array--this means the beam-scanning is mostly in the vertical orientation--very little ability to perform horizontal beam-forming).
Now, examine the 18 inch cube with 20,000 antennas: They will conceivably place transmitters along the all 6 sides. Each side has a surface area of roughly 45 x 45 cm and will contain roughly 3,300 antennas. Assuming the antennas are miniaturized such that each antenna length is ~ 1 cm, then each side of the cube can support roughly 256 antennas with optimal mutual coupling. If each side were to contain over 3,000 antennas, the separation would decrease to almost zero, resulting in mutual coupling close to 100%; thereby rendering the system useless.
Note: All the antennas would need to be placed along the surface of the cube, if more antennas were placed inside, the transmitted power from the inner antennas would just get absorbed by the outer layers.