Critical exponent and asymptotic critical exponent of the generalized Allouche–Johnson words

Prove that, for every integer k >= 1, the infinite word x_k defined by the locally catenative recurrence has critical exponent k+1, attained precisely by the words 0^(k+1) and 1^(k+1), contains no factor of length 2n+k and period n, and consequently has asymptotic critical exponent 2.

Background

The paper introduces a family of infinite words x_k generated by locally catenative recurrences, with the Thue–Morse word corresponding to k=1, the Fibonacci–Thue–Morse word to k=2, and the Allouche–Johnson word to k=3. For the first three cases, the paper cites or proves increasingly strong restrictions on repetitions: the Thue–Morse word is overlap-free, the Fibonacci–Thue–Morse word contains no factor of length 2n+2 and period n, and the Allouche–Johnson word contains no factor of length 2n+3 and period n.

The conjecture proposes a uniform formula for the critical exponent and a uniform bound on repetitions throughout the entire family. Establishing it would generalize the paper’s results for k=1, 2, and 3 to all k >= 1.