Critical exponent conjecture for the generalized Thue–Morse words

For each integer $k\geq 1$, let ${\bf x}_k$ be the infinite word defined as the limit of the locally catenative sequence described above. The critical exponent conjecture for ${\bf x}_k$. The infinite word ${\bf x}_k$ 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 therefore has asymptotic critical exponent $2$. This extends the known overlap-free and related properties of the cases $k=1$, $k=2$, and $k=3$; the assertion for all $k\geq 1$ remains open.