Powers of 2 not containing Digit Power of 2/Historical Note

Historical Note on Powers of 2 not containing Digit Power of 2
This question appears to have first been raised by in $1989$.

While no formal attempt has been made to solve it, several searches were made in response, as follows:
 * verified the result up to $2^{167}$.
 * showed that if the powers of $2$ contain between $500$ and $1000$ digits, the digits $1, 2, 4, 8$ occur fairly normally.
 * checked the powers of $2$ up to $2^{3320}$, finding no other solution.
 * searched through $2^{20703}$, also finding no other solution.
 * Finally, went to $2^{31000}$, which contains $9332$ digits, with the same result.