How plausible is the claim about a 200 kW battery solution?
203kW / 36 fans = 5.6kW per fan. 

Working voltage of 38V implies 10S Lipo (3.8V per cell).

5.6kW / 38V = 150A. We want 3 minutes at full power. At half power it will draw 75A for 6 minutes (max duration). A battery capacity of greater than 150*(3/60) or 75*(6/60) = 7.5Ah per fan will be required.

Can it be done? 

Looks like 120mm diameter fans will fit in the space provided. Here's a 120mm fan that weighs 1kg and produces 7.5kg thrust on 12S:-

120mm 11 Blade Alloy EDF 700kv - 7000watt

On 10S it would draw about 30% less power and produce about 15% less thrust, so let's say 5kW and 6.5kg (the fans they are using may have different motors, but we can expect similar performance at the same power level).   

And here's a 10S 4Ah battery that weighs 905g:-     

ZIPPY Compact 4000mAh 10S 25C Lipo Pack

The board appears to use a total of 72 batteries - two batteries per fan. 2 x 4Ah = 8Ah, close to our required capacity. Max discharge rate is 4 x 25C = 100A per battery or 200A per parallel pair (and we 'only' need 150A!). Max charge rate is 5C, well above the 2C rate required for a 35 minute charge. At $67 per pack the total battery cost is $4824.

Our 72 batteries weigh 905g x 72 = 65kg. The 36 fans weigh 36kg. Add another 10% for ESCs, wiring and support structure, and we get a total board weight of ~110kg. This board should generate 6.5kg x 36 = 234kg thrust in free air. At half power thrust would be reduced to about 75%, but could be boosted by ground effect - so perhaps 210kg of 'duration' hovering thrust. Take away the weight of the board and you have a payload capacity of 100kg.  

Looks possible!