Powers of 2 whose Digits are Powers of 2

Open Question
The only known powers of $2$ whose digits are also all powers of $2$ are:
 * $1, 2, 4, 8, 128$

Are there any more?

Progress
Demonstrated up to at least $2^{10 \, 000 \, 000}$.

This can be achieved by looking at the lowest $20$ digits only, by calculating those of successive powers of $2$ after applying the modulo operation.