Why do a few of the LEDs in my lamp stay on at very dim light?
This is slightly different than most of the LED-style questions asking about the dim glow but has the same cause, most likely.
The remote may be using a very small amount of current to maintain its functionality.  This small amount is enough to make some LED’s glow.
To eliminate the glow, a different remote that is compatible with LED’s will be needed, or in some cases, an incandescent lamp in one of the positions will do the trick.
We can not detect the tiny amount of power being used with an incandescent, but the LED’s are so efficient, they glow because of the remote power usage.