Just looked for flights to Asia on CX and got them for just over £3.4k including legs to Singapore and back from Jakarta for 2 weeks time (rest of Asia is common rated with HK so if you need to fly internally worth booking as 1 ticket). Absolute lowest fare you can get is about £3.2k with CX at the moment (possible sale fares could be lower if and when they are released). Have also found for another reader LX for under £2k to HKG in August but otherwise you are right, Finnair can have good fares.
Also worth looking at AF to Guangzhou if that is useful as that is less, and TK to HKG or CAN can also be good value. I would expect EK/QR to release some lower fares at some point soon as they have removed “O” Class Business fares (Emirates) which are there specials so these could come back in the next few days/weeks.