They don't say how soon, but this is encouraging news for those who don't mind public transport to/from the airport. The current option is a bit more involved than I like to bother with and have taken a taxi, and for those 6 am flights home a taxi will still be my preference.
And today they announced it starts 7/8 and will cost 900 HUF, about $3.50 - 4.00 US.
https://welovebudapest.com/budapest.and.hungary/a.new.direct.airport.bus.will.soon.serve.central.budapest