I have a K3. It has the Kärcher standard connection and fits. If the K4 also has this standard, it will certainly fit. According to other questions/answers, K5 also works. So K4 should not be a problem either.

I have a K2 and it fits perfectly. Since the compatibility list also includes K4 models (although not the Full Control), I assume it should work. There are also nozzles included where you can adapt it to stronger models.