Newbie here. I've been messing with this issue a bit as well and haven't really made much progress. If it's any help here's my findings after doing the firmware update to v1.5 (v1.40 addressed this issue - well apparently it did). Granted I did this on the K-1 mark 1 instead of K-1 II, but the difference I did see seems negligible, and if anything worse. Settings were 2min exposure at ISO3200 with the lens cap on: I did this purely to test the before and after difference. I've had worse issues with this than seen below when doing actual star photography. Maybe the cold has an effect? Anywho, 1:1 crops:
Firmware v1.0
Firmware v1.5
Firmware v1.5 after pixel mapping