If 4k is four times the pixel count of 1080, then 2k means 1440 (-ish, it should be 1530) - that’s fine. But then 8k must be 3050, but it is actually 4320!!!
So it can not refer to the number of pixels (quadratic scaling). On the other hand, if we assume linear scaling and 8k is 4320 and 4k is 2160, then 2k is 1080 - but 2k is never used in that context!
Edit: as you can see I’m very passionate about this XD