Pseudoperiodic word avoidance conjecture for binary words

Let $(a,b)$ be a pair of positive integers with $1\leq a<b$. An infinite binary word has pseudoperiod $(a,b)$ if each position is compatible with a repetition having periods $a$ and $b$; a word is avoiding $3^+$-powers if it contains no factor whose exponent is at least $3$. Pseudoperiodic word avoidance conjecture. For all pairs $(a,b)$ with $1\leq a<b$ and $b\ne 2a$, there exists an infinite binary word with pseudoperiod $(a,b)$ and avoiding $3^+$-powers. The conjecture is motivated by the observed critical exponents in the computed examples, all of which are at most $3^+$; its general case remains open.