So, if you carry 2.5x the battery you need, so a 40% charge is all you ever need, then maybe? Although obviously, that would add to your weight, reducing your range.
If you were willing to hook up a supply of cooling water while charging, you might be able to do somewhat better - one of the key limiting factors in fast charging is the batteries heating up.
[1] https://insideevs.com/news/356710/tesla-model-3-test-v3-supe...
https://web.mit.edu/evt/summary_battery_specifications.pdf
https://batteryuniversity.com/index.php/learn/article/what_i...
And generally bigger battery packs can charge at higher kilowatts. Here's an electric ferry with a 4.1 MWh battery which can charge at up 6 megawatts. 600 amps at 10,000 volts:
https://www.youtube.com/watch?v=rE_M1n-ClOA
So, like the ferry, these aircraft probably have big batteries that they charge enough to get to the next destination with a buffer for emergencies. The closer you get to 100% state of charge the slower the system will charge in order to protect the battery.
If I had to guess the 5 to 7 minute figure is based on some theoretical magical battery that nobody can buy yet. Something with Wh/density much greater than the current best lithium ion (1000 to 1200Wh/kg vs 245Wh/kg) and capable of much higher C rates for charge and discharge. It's a mock up so they don't have to, or want to share any more detail than a optimistic time figure.
The article is light on specifics but this stood out to me, most Tesla's take longer to charge. I'd imagine an aircraft would need more power than a Tesla, so how are they transferring energy so fast that automobiles can't do it but aircraft can?