Shallit's critical-exponent conjecture for Thue-Morse-like sequences

Let $k\geq 1$, and let $\mathbf{x}_k$ be the Thue-Morse-like sequence described in the source. The critical exponent is the supremum of the exponents of its finite factors, while the asymptotic critical exponent concerns repetitions whose lengths tend to infinity. Shallit's critical-exponent conjecture. The sequence $\mathbf{x}_k$ has critical exponent

$$k+1,$$

attained by the words $0^{k+1}$ and $1^{k+1}$. It contains no factor of length $2n+k$ and period $n$, and consequently its asymptotic critical exponent is

$$2.$$

This is the second conjecture formulated by Shallit for the family of Thue-Morse-like sequences, alongside the factor-complexity conjecture. The source gives no resolution, so the conjecture remains open.