Posting here in case someone else wants to know if Logitech mk240 works with Xim4. It does! Yesterday
I bought one to replace my wired keyboard and go wireless. I just plugged that old-style (it's large) receiver instead of my wired keyboard and voila! Also, no problems with keys rollover.
Never tried the mouse which comes with the keyboard. From what I know the mouse it won't work with Xim4 together with the keyboard because they both share the same receiver.