Erdős’s restricted-digit question for powers of two

Determine whether every power of 2 other than 1, 4, and 256 has at least one base-3 digit equal to 2.

Background

The problem concerns the interaction between representations of integers in bases 2 and 3. A counterexample would be a power of 2 whose base-3 expansion uses only the digits 0 and 1. The paper reports that Dimitrov and Hove established that any counterexample would have to be structurally large, specifically requiring a representation as a sum of at least 25 distinct powers of 3, but does not state that the question has been resolved.